PHYSUCS IN MY GENERATION

105170

3!

PHYSICS IN
MY GENERATION

PHYSICS IN
MY GENERATION

A selection of papers

MAX BORN

F.R.S., N.L.

PERGAMON PRESS
London &. New YorA

First published 1956

Printed in England by

ADLARD AND SON LIMITED

London and Dorking

Published in Great Britain by Pergmon Press,
4 6? 5 Fit&oy Square, London W.i, and 122 East
Street, New Tork 22, N.Y.

CONTENTS

PAGE

Preface vii

Introduction to Einstein's Theory of Relativity (1921) . . i
Physical Aspects of Quantum Mechanics .... 6

On the Significance of Collision Processes in the Under-
standing of Quantum Mechanics . . . .14

On the Meaning of Physical Theories . . . 1 7

Some Philosophical Aspects of Modern Physics , . 37

Cause, Purpose, and Economy in Natural Laws . . . 55

Einstein's Statistical Theories 80

Physics and Metaphysics ....... 93

Physics in the Last Fifty Years . . . . . . 109

The Conceptual Situation in Physics and the Prospects of

its Future Development . . . . . .123

The Interpretation of Quantum Mechanics . . . 140

Physical Reality 151

Is Classical Mechanics in fact Deterministic ? . . .164
Astronomical Recollections . . . . . .171

Statistical Interpretation of Quantum Mechanics . . 177

Physics and Relativity 189

Development and Essence of the Atomic Age . , . 207

A New Year's Message 223

From the Postscript to The Restless Universe (1951) . * 225

TO

SIR EDWARD APPLETON

G.B.E.,K.C.B.,F.R.S.,N.L.

Vice-Chancellor and Principal
of Edinburgh University

PREFACE

THE idea of collecting these essays occurred to me when, in the
leisure of retirement, I scanned some of my own books and found
that two of the more widely read show a startling change of attitude
to some of the fundamental concepts of science. These are
Einstein 9 s Theory of Relativity of 1921 and the American edition of
The Restless Universe of 1951. I have taken the introduction of the
former as the first item of this collection, the postscript to the latter
as its last. These books agree in the relativistic concept of space
and time, but differ in many other fundamental notions. In 1921
I believed and I shared this belief with most of my contemporary
physicists that science produced an objective knowledge of the
world, which is governed by deterministic laws. The scientific
method seemed to me superior to other, more subjective ways of
forming a picture of the world philosophy, poetry, and religion;
and I even thought the unambiguous language of science to be a
step towards a better understanding between human beings.

In 1951 I believed in none of these things. The border between
object and subject had been blurred, deterministic laws had been
replaced by statistical ones, and although physicists understood
one another well enough across all national frontiers they had
contributed nothing to a better understanding of nations, but had
helped in inventing and applying the most horrible weapons of
destruction.

I now regard my former belief in the superiority of science over
other forms of human thought and behaviour as a self-deception
due to youthful enthusiasm over the clarity of scientific thinking
as compared with the vagueness of metaphysical systems.

Still, I believe that the rapid change of fundamental concepts
and the failure to improve the moral standards of human society
are no demonstration of the uselessness of science in the search for
truth and for a better life.

The change of ideas was not arbitrary, but was forced on the
physicists by their observations. The final criterion of truth is the
agreement of a theory with experience, and it is only when all
attempts to describe the facts in the frame of accepted ideas fail
that new notions are formed, at first cautiously and reluctantly,
and then, if they are experimentally confirmed, with increasing

Vlii PREFACE

confidence. In this way the classical philosophy of science was
transformed into the modern one, which culminates in NIELS
BOHR'S Principle of Complementarity.

To illustrate this process I have selected some of my popular
writings covering the period of 30 years which lies between the
publication dates of the books mentioned above, and have framed
them by the introduction to the first and the postscript to the
second. Some of the articles are only loosely connected with the
main theme, such as one on the minimum principles in physics,
several discussions of EINSTEIN'S work, and a modest attempt at
autobiography. The remaining articles deal with the philosophical
background of physics and its revolutionary changes during my
lifetime. There are many repetitions which could not be avoided
without spoiling the inner structure of the articles; but I think
that each treatment of a problem illuminates it from a different
angle, though all of them are given from my personal point of
view. The articles are ordered chronologically.

I hope that the collection may transmit to the reader something
of the adventurous spirit of a great period of physics.

I am very much indebted to Dr. D. J. HOOTON for helping me in
reading the proofs, and to the staff of the publishing firm for their
willingness to comply with my wishes, and for the excellent printing.

ACKNOWLEDGMENT

The author, and the publishers, would like to express
thanks to those concerned for permission to reproduce
material first published elsewhere. In every case
details of first publications are given at the head of
each article.

INTRODUCTION TO "EINSTEIN'S THEORY
OF RELATIVITY" (1921)

HPHE world is not presented to the reflective mind as a finished
* product. The mind has to form its picture from innumerable
sensations, experiences, communications, memories, perceptions.
Hence there are probably not two thinking people whose picture
of the world coincides in every respect.

When an idea in its main lines becomes the common property
of large numbers of people, the movements of spirit that are called
religious creeds, philosophic schools, and scientific systems arise;
they present the aspect of a chaos of opinions, of articles of faith,
of convictions, that resist all efforts to disentangle them. It seems
a sheer impossibility to find a thread that will guide us along a
definite path through these widely ramified doctrines that branch
off perchance to recombine at other points.

What place are we to assign to EINSTEIN'S theory of relativity,
of which this book seeks to give an account? Is it only a special
part of physics or astronomy, interesting in itself but of no great
importance for the development of the human spirit? Or is it at
least a symbol of a particular trend of thought characteristic of
our times? Or does it itself, indeed, signify a 'world-view*
(Weltanschauung) ? We shall be able to answer these questions
with confidence only when we have become acquainted with the
content of EINSTEIN'S doctrine. But we may be allowed to present
here a point of view which, even if only roughly, classifies the
totality of all world-views and ascribes to EINSTEIN'S theory a
definite position within a uniform view of the world as a whole.
The world is composed of the ego and the non-ego, the inner
world and the outer world. The relations of these two poles are
the object of every religion, of every philosophy. But the part
that each doctrine assigns to the ego in the world is different.
The importance of the ego in the world-picture seems to me a
measure according to which we may order confessions of faith,
philosophic systems, world-views rooted in art or science, like
pearls on a string. However enticing it may be to pursue this
idea through the history of thought, we must not diverge too far
from our theme, and we shall apply it only to that special realm of
human thought to which EINSTEIN'S theory belongs to natural
science.

2 INTRODUCTION TO EINSTEIN'S THEORY OF RELATIVITY

Natural science is situated at the end of this series, at the point
where the ego, the subject, plays only an insignificant part; every
advance in the mouldings of the concepts of physics, astronomy
and chemistry denotes a further step towards the goal of excluding
the ego. This does not, of course, deal with the act of knowing,
which is bound to the subject, but with the finished picture of
Nature, the basis of which is the idea that the ordinary world
exists independently of and uninfluenced by the process of knowing.

The doors through which Nature imposes her presence on us
are the senses. Their properties determine the extent of what is
accessible to sensation or to intuitive perception. The further we
go back in the history of the sciences, the more we find the natural
picture of the world determined by the qualities of sense. Older
physics was subdivided into mechanics, acoustics, optics and theory
of heat. We see the connexions with the organs of sense, the per-
ceptions of motion, impressions of sound, light, and heat. Here
the qualities of the subject are still decisive for the formation of
concepts. The development of the exact sciences leads along
a definite path from this state to a goal which, even if far from being
attained, yet lies clearly exposed before us: it is that of creating a
picture of nature which, confined within no limits of possible
perception or intuition, represents a pure structure of concepts,
conceived for the purpose of depicting the sum of all experiences
uniformly and without inconsistencies.

Nowadays mechanical force is an abstraction which has only
its name in common with the subjective feeling of force. Mech-
anical mass is no longer an attribute of tangible bodies but is also
possessed by empty spaces filled only by ether radiation. The
realm of audible tones has become a small province in the world
of inaudible vibrations, distinguishable physically from these
solely by the accidental property of the human ear which makes it
react only to a definite interval of frequency numbers. Modern
optics is a special chapter out of the theory of electricity and
magnetism, and it treats of the electro-magnetic vibrations of all
wave-lengths, passing from the shortest y-rays of radioactive sub-
stances (having a wavelength of one hundred millionth of a milli-
metre) over the X- (Rontgen) rays, the ultraviolet, visible light, the
infra-red, to the longest wireless (Hertzian) waves (which have a
wave-length of many kilometres). In the flood of invisible light
that is accessible to the mental eye of the physicist, the material
eye is almost blind, so small is the interval of vibrations which it
converts into sensations. The theory of heat, too, is but a special
part of mechanics and electrodynamics. Its fundamental concepts

INTRODUCTION TO EINSTEIN'S THEORY OF RELATIVITY 3

of absolute temperature, of energy, and of entropy belong to the
most subtle logical constructions of exact science, and, again, only
their name still carries a memory of the subjective impression of
heat or cold.

Inaudible tones, invisible light, imperceptible heat, these con-
stitute the world of physics, cold and dead for him who wishes to
experience living Nature, to grasp its relationships as a harmony,
to marvel at her greatness in reverential awe. GOETHE abhorred
this motionless world. His bitter polemic against NEWTON, whom
he regarded as the personification of a hostile view of Nature,
proves that it was not merely a question of an isolated struggle
between two investigators about individual questions of the theory
of colour. GOETHE is the representative of a world-view which is
situated somewhere near the opposite end of the scale suggested
above (constructed according to the relative importance of the
ego), that is, the end opposite to that occupied by the world-picture
of the exact sciences. The essence of poetry is inspiration, in-
tuition, the visionary comprehension of the world of sense in symbolic
forms. The source of poetry is personal experience, whether it be
the clearly conscious perception of a sense-stimulus, or the power-
fully represented idea of a relationship or connexion. What is
logically formal and rational plays no part in the world-picture of
such a type of gifted or indeed heaven-blessed spirit. The world
as the sum of abstractions that are connected only indirectly with
experience is a province that is foreign to it. Only what is directly
presented to the ego, only what can be felt or at least represented
as a possible experience is real to it and has significance for it.
Thus to later readers, who survey the development of exact methods
during the century after GOETHE'S time and who measure the
power and significance of GOETHE'S works on the history of natural
science by their fruits, these works appear as documents of a
visionary mind, as the expression of a marvellous sense of one-ness
with (Einfiihlung) the natural relationships, but his physical
assertions will seem to such a reader as misunderstandings and
fruitless rebellions against a greater power, whose victory was
assured even at that time.

Now in what does this power consist, what is its aim and device?

It both takes and renounces. The exact sciences presume to
aim at making objective statements, but they surrender their
absolute validity. This formula is to bring out the following
contrast.

All direct experiences lead to statements which must be allowed
a certain degree of absolute validity. If I see a red flower, if I

4 INTRODUCTION TO EINSTEIN'S THEORY OF RELATIVITY

experience pleasure or pain, I experience events which it is meaning-
less to doubt. They are indubitably valid, but only for me. They
are absolute, but they are subjective. All seekers after human
knowledge aim at taking us out of the narrow circle of the ego,
out of the still narrower circle of the ego that is bound to a moment
of time, and at establishing common ground with other thinking
creatures. First a link is established with the ego as it is at another
moment, and then with other human beings or gods. All religions,
philosophies, and sciences have been evolved for the purpose of
expanding the ego to the wider community that e we 9 represent.
But the ways of doing this are different. We are again confronted
by the chaos of contradictory doctrines and opinions. Yet we no
longer feel consternation, but order them according to the import-
ance that is given to the subject in the mode of comprehension
aimed at. This brings us back to our initial principle, for the
completed process of comprehension is the world-picture. Here
again the opposite poles appear. The minds of one group do not
wish to deny or to sacrifice the absolute, and they therefore remain
clinging to the ego. They create a world-picture that can be
produced by no systematic process, but by the unfathomable action
of religious, artistic, or poetic means of expression in other souls.
Here faith, pious ardour, love of brotherly communion, but often
also fanaticism, intolerance, intellectual suppression hold sway.

The minds of the opposite group sacrifice the absolute. They
discover often with feelings of terror the fact that inner experi-
ences cannot be communicated. They no longer fight for what
cannot be attained, and they resign themselves. But they wish to
reach agreement at least in the sphere of the attainable. They
therefore seek to discover what is common in their ego and in that
of the other egos; and the best that was there found was not the
experiences of the soul itself, not sensations, ideas, or feelings, but
abstract concepts of the simplest kind numbers, logical forms;
in short, the means of expression of the exact sciences. Here we
are no longer concerned with what is absolute. The height of a
cathedral does not, in the special sphere of the scientist, inspire
reverence, but is measured in metres and centimetres. The course
of life is no longer experienced as the running out of the sands of
time, but is counted in years and days. Relative measures take
the place of absolute impressions. And we get a world, narrow,
one-sided, with sharp edges, bare of all sensual attraction, of all
colours and tones. But in one respect it is superior to other world-
pictures: the fact that it establishes a bridge from mind to mind
cannot be doubted. It is possible to agree as to whether iron has a

INTRODUCTION TO EINSTEIN'S THEORY OF RELATIVITY 5

specific gravity greater than wood, whether water freezes more
readily than mercury, whether Sirius is a planet or a star. There
may be dissensions, it may sometimes seem as if a new doctrine
upsets all the old facts, yet he who has not shrunk from the effort
of penetrating into the interior of this world will feel that the
regions known with certainty are growing, and this feeling relieves
the pain which arises from solitude of the spirit, and the bridge to
kindred spirits becomes built.

We have endeavoured in this way to express the nature of
scientific research, and now we can assign EINSTEIN'S theory of
relativity to its category.

In the first place, it is a pure product of the striving after the
liberation of the ego, after the release from sensation and perception.
We spoke of the inaudible tones, of the invisible light, of physics.
We find similar conditions in related sciences, in chemistry which
asserts the existence of certain (radioactive) substances, of which
no one has ever perceived the smallest trace with any sense directly
or in astronomy, to which we refer below. These Extensions
of the world', as we might call them, essentially concern sense-
qualities. But everything takes place in the space and the time
which was presented to mechanics by its founder, NEWTON. Now,
EINSTEIN'S discovery is that this space and this time are still entirely
embedded in the ego, and that the world-picture of natural science
becomes more beautiful and grander if these fundamental con-
cepts are also subjected to relativization. Whereas, before, space
was closely associated with the subjective, absolute sensation of
extension, and time with that of the course of life, they are now
purely conceptual schemes, just as far removed from direct per-
ception as entities, as the whole region of wave-lengths of present-
day optics is inaccessible to the sensation of light except for a very
small interval. But just as in the latter case, the space and time of
perception allow themselves to be ordered, without giving rise to
difficulties, into the system of physical concepts. Thus an
objectivation is attained, which has manifested its power by pre-
dicting natural phenomena in a truly wonderful way. We shall
have to speak of this in detail in the sequel.

Thus the achievement of EINSTEIN'S theory is the relativization
and objectivation of the concepts of space and time. At the
present day it is the final picture of the world as presented by
science.

PHYSICAL ASPECTS OF QUANTUM
MECHANICS*

[First published in Nature, Vol. 119, pp. 354~357> 1927-]

""pHE purpose of this communication is not to give a report on the
A present status of quantum mechanics. Such a report has recently
been published by W. HEISENBERG, the founder of the new theory
(Die Naturwissenschaften, 45, 989, 1926). Here we shall make an
attempt to understand the physical significance of the quantum
theoretical formulae.

At present we have a surprisingly serviceable and adaptable
apparatus for the solution of quantum theoretical problems. We
must insist here that the different formulations, the matrix theory,
DIRAC'S non-commutative algebra, SGHRODINGER'S partial differen-
tial equations, are mathematically equivalent to each other, and
form, as a whole, a single theory. This theory enables us to compute
the stationary states of atoms and the corresponding radiation, if we
neglect the reaction of the radiation on the atoms; it would seem
that in this respect we have nothing more to wish for, since the
result of every example in which the calculations are carried out
agrees with experiment.

This question, however, of the possible states of matter does not
exhaust the field of physical problems. Perhaps more important still
is the question of the course of the phenomena that occurs when
equilibrium is disturbed. Classical physics was entirely concerned
with this question, as it was almost powerless toward the problem
of structure. Conversely, the question of the course of phenomena
practically disappeared from the quantum mechanics, because it
did not immediately fit into the formal developments of the theory.
Here we shall consider some attempts to treat this problem on the
new mechanics.

In classical dynamics the knowledge of the state of a closed system
(the position and velocity of all its particles) at any instant deter-
mines unambiguously the future motion of the system; that is the
form that the principle of causality takes in physics. Mathematically,

* Extension of a paper read before Section A (Mathematics and Physics) of the
British Association at Oxford on August loth, 1926. Translated by Mr. ROBERT
OPPENHEIMER. The author is very much obliged to Mr. OPPENHEIMER for his
careful translation.

PHYSICAL ASPECTS OF QUANTUM MECHANICS 7

this is expressed by the fact that physical quantities satisfy differen-
tial equations of a certain type. But besides these causal laws,
classical physics always made use of certain statistical considerations.
As a matter of fact, the occurrence of probabilities was justified by
the fact that the initial state was never exactly known; so long as
this was the case, statistical methods might be, more or less pro-
visionally, adopted.

The elementary theory of probability starts with the assumption
that one may with reason consider certain cases equally probable,
and derives from this the probability of complicated combinations
of these. More generally: starting with an assumed distribution (for
example, a uniform one, with equally probable cases) a dependent
distribution is derived. The case in which the derived distribution
is entirely or partly independent of the assumed initial distribution
is naturally particularly important.

The physical procedure corresponds to this: we make an assump-
tion about the initial distribution, if possible, one about equally
probable cases, and we then try to show that our initial distribution
is irrelevant for the final, observable, results. We see both parts of
this procedure in statistical mechanics: we divide the phase space
into equally probable cells, guided only by certain general theorems
(conservation of energy, LIOUVILLE'S theorem) ; at the same time
we try to translate the resulting space-distribution into a distribu-
tion in time. But the ergodic hypothesis, which was to effect this
translation, and states that every system if left to itself covers in
time its phase space uniformly, is a pure hypothesis and is likely to
remain one. It thus seems that the justification of the choice of
equally probable cases by dividing the phase space into cells can
only be derived a posteriori from its success in explaining the observed
phenomena.

We have a similar situation in all cases where considerations of
probability are used in physics. Let us take as an example an atomic
collision the collision of an electron with an atom. If the kinetic
energy of the electron is less than the first excitation potential of
the atom the collision is elastic: the electron loses no energy. We
can then ask in what direction the electron is deflected by the
collision. The classical theory regards each such collision as causally
determined. If one knew the exact position and velocity of all the
electrons in the atom and of the colliding electron, one could com-
pute the deflexion in advance. But unfortunately we again lack this
information about the details of the system; we have again to be
satisfied with averages. It is usually forgotten that in order to obtain
these, we have to make an assumption about equally probable

8 PHYSICAL ASPECTS OF QUANTUM MECHANICS

configurations. This we do in the most 'natural 5 way by expressing
the co-ordinates of the electron in its initial path (relative to the
nucleus) in terms of angle variables and phases, and by treating
equal phase intervals as equally probable. But this is only an
assumption, and can only be justified by its results.

The peculiarity of this procedure is that the microscopic co-
ordinates are only introduced to keep the individual phenomena at
least theoretically determinate. For practical purposes they do not
exist: the experimentalist only counts the number of particles
deflected through a given angle, without bothering about the
details of the path; the essential part of the path, in which the
reaction of the atom on the electron occurs, is not open to observa-
tion. But from such numerical data we can draw conclusions about
the mechanism of the collision. A famous example of this is the work
of RUTHERFORD on the dispersion of a-particles; here, however, the
microscopic co-ordinates are not electronic phases, but the distance
of the nucleus from the original path of the a-particle. From the
statistics of the dispersion, RUTHERFORD could prove the validity of
COULOMB'S law for the reaction between the nucleus and the a-
particle. The microscopic co-ordinate had been eliminated from the
theoretical formula for the distribution of the particles over different
angles of deflexion.

We thus have an example of the evaluation of a field of force by
counting, by statistical methods, and not by the measurement of an
acceleration and NEWTON'S second law.

This method is fundamentally like that which makes us suspect
that a dice is false if one face keeps turning up much more often than
every sixth throw; statistical considerations indicate a torque.
Another example of this is the 'barometer formula'. Of course, we
can derive this dynamically, if we regard the air as a continuum
and require equilibrium between hydrodynamical pressure and
gravity; but actually pressure is only defined statistically as the
average transport of momentum in the collisions of the molecules,
and it is therefore not merely permissible but also fundamentally
more sound to regard the barometer formula as a counting of the
molecules in a gravitational field, from which the laws of the field
may be derived.

These considerations were to lead us to the idea that we could
replace the Newtonian definition of force by a statistical one. Just
as in classical mechanics we concluded that there was no external
force acting if the motion of the particle was rectilinear, so here we
should do so if an assembly of particles was uniformly distributed
over a range. (The choice of suitable co-ordinates leads to similar

PHYSICAL ASPECTS OF QUANTUM MECHANICS 9

problems on both theories.) The magnitude of a force, classically
measured by the acceleration of a particle, would here be measured
by the inhomogeneity of an assembly of particles.

In the classical theory we are of course faced with the problem
of reducing the two definitions offeree to one, and that is the object
of all attempts at a rational foundation of statistical mechanics; we
have tried to make clear, though, that these have not been altogether
successful, because in the end the choice of equally probable cases
cannot be avoided.

With this preparation we turn our attention to quantum mech-
anics. It is notable that here, even historically, the concept of a priori
probability has played a part that could not be thrown back on
equally probable cases, for example, in the transition-probabilities
for emission. Of course this might be merely a weakness of the theory.

It is more important that formal quantum mechanics obviously
provides no means for the determination of the position of particles
in space and time. One might object that according to SCHRODINGER
a particle cannot have any sharply defined position, since it is only a
group of waves with vague limits; but I should like to leave aside this
notion of 'wave-packets', which has not been, and probably cannot
be, carried through. For SCHRODINGER'S waves move not in ordinary
space but in configuration space, that has as many dimensions as
the degrees of freedom of the system (3JV for JV particles). The
quantum theoretical description of the system contains certain
declarations about the energy, the momenta, the angular momenta of
the system; but it does not answer, or at least only answers in the
limiting case of classical mechanics, the question of where a certain
particle is at a given time. In this respect the quantum theory is in
agreement with the experimentalists, for whom microscopic co-
ordinates are also out of reach, and who therefore only count
instances and indulge in statistics. This suggests that quantum
mechanics similarly only answers properly-put statistical questions,
and says nothing about the course of individual phenomena. It
would then be a singular fusion of mechanics and statistics.

According to this, we should have to connect with the wave-
equations such a picture as this: the waves satisfying this equation
do not represent the motion of particles of matter at all; they only
determine the possible motions, or rather states, of the matter.
Matter can always be visualised as consisting of point masses
(electrons, protons), but in many cases the particles are not to be
identified as individuals, e.g. when these form an atomic system.
Such an atomic system has a discrete set of states; but it also has a
continuous range of them, and these have the remarkable property

io PHYSICAL ASPECTS OF QUANTUM MECHANICS

that in them a disturbance is propagated along a path away from
the atom, and with finite velocity, just as if a particle were being
thrown out. This fact justifies, even demands, the existence of
particles, although this cannot, in some cases as we have said, be
taken too literally. There are electromagnetic forces between these
particles (we neglect for the moment the finite velocity of propaga-
tion); they are, so far as we know, given by classical electro-
dynamics in terms of the positions of the particles (for example, a
Coulomb attraction). But these forces do not, as they did classically,
cause accelerations of the particles; they have no direct bearing on
the motion of the particles. As intermediary there is the wave field:
the forces determine the vibrations of a certain function i/r that
depends on the positions of all the particles (a function in configura-
tion space), and determine them because the coefficients of the
differential equation for t/r involve the forces themselves.

A knowledge of i/r enables us to follow the course of a physical
process in so far as it is quantum-mechanically determinate: not in
a causal sense but in a statistical one. Every process consists of
elementary processes, which we are accustomed to call transitions
or jumps; the jump itself seems to defy all attempts to visualize it,
and only its result can be ascertained. This result is, that after the
jump, the system is in a different quantum state. The function i/r
determines these transitions in the following way: every state of the
system corresponds to a particular characteristic solution, an
Eigenfunktiori) of the differential equation; for example, the normal
state the function i/f 1} the next state ^ 2 , etc. For simplicity we assume
that the system was originally in the normal state; after the occur-
rence of an elementary process the solution has been transformed
into one of the form

which represents a superposition of a number of eigenfunktions with
definite amplitudes c lt c z , C B , . . . Then the squares of the amplitudes
^i 2 j ^a 2 > gi ve the probability that after the jump the system is
in the i, 2, 3, state. Thus c^ is the probability that in spite of the
perturbation the system remains in the normal state, 2 2 the prob-
ability that it has jumped to the second, and so on.* These prob-
abilities are thus dynamically determined. But what the system
actually does is not determined, at least not by the laws that are at

* We may point out that this theory is not equivalent to that of BOHR, KRAMERS,
and SLATER. In the latter the conservation of energy and momentum are
purely statistical laws; on the quantum theory their exact validity follows from
the fundamental equations.

PHYSICAL ASPECTS of QUANTUM MECHANICS n

present known. But this is nothing new, for we saw above that the
classical theory for example, for the collision problem only gave
probabilities. The classical theory introduces the microscopic co-
ordinates which determine the individual process, only to eliminate
them because of ignorance by averaging over their values; whereas
the new theory gets the same results without introducing them at all.
Of course, it is not forbidden to believe in the existence of these co-
ordinates ; but they will only be of physical significance when methods
have been devised for their experimental observation.

This is not the place to consider the associated philosophical
problems; we shall only sketch the point of view which is forced
upon us by the whole of physical evidence. We free forces of their
classical duty of determining directly the motion of particles and
allow them instead to determine the probability of states. Whereas
before it was our purpose to make these two definitions of force
equivalent, this problem has now no longer, strictly speaking, any
sense. The only question is why the classical definition is so useful
for a large class of phenomena. As always in such cases, the answer
is : because the classical theory is a limiting case of the new one.
Actually, it is usually the c adiabatic' case with which we have to
do: i.e. the limiting case where the external force (or the reaction
of the parts of the system on each other) acts very slowly. In this
case, to a very high approximation

that is, there is no probability for a transition, and the system is in
the initial state again after the cessation of the perturbation. Such a
slow perturbation is therefore reversible, as it is classically. One can
extend this to the case where the final system is really under different
conditions from the initial one; i.e. where the state has changed
adiabatically, without transition. That is the limiting case with which
classical mechanics is concerned.

It is, of course, still an open question whether these conceptions
can in all cases be preserved. The problem of collisions was with
their help given a quantum mechanical formulation ; and the result
is qualitatively in full agreement with experiment. We have here a
precise interpretation of just those observations which may be
regarded as the most immediate proof of the quantized structure of
energy, namely, the critical potentials, that were first observed by
FRANCK and HERTZ. This abrupt occurrence of excited states with
increasing electronic velocity of the colliding electron follows
directly out of the theory. The theory, moreover, yields general
formulae for the distribution of electrons over the different angles of

12 PHYSICAL ASPECTS OF QUANTUM MECHANICS

deflexion, that differ in a characteristic way from the results that
we should have expected classically. This was first pointed out by
W. ELSASSER (Die Naturwissenschqften, Vol. 13, p. 711, 1925) before
the development of the general theory. He started with DE BROGUE'S
idea that the motion of particles is accompanied by waves, the
frequency and wave-length of which is determined by the energy
and momentum of the particle. ELSASSER computed the wave-length
for slow electrons, and found it to be of the order of io~ 8 cm., which
is just the range of atomic diameters. From this he concluded that
the collision of an electron with an atom should give rise to a
diffraction of the DE BROGLIE waves, rather like that of light which
is scattered by small particles. The fluctuation of the intensities in
different directions would then represent the irregularities in the
distribution of the deflected electrons. Indications of such an effect
are given by the experiments of DAVISSON and KUNSMANN (Phys. Rev.,
Vol. 22, p. 243, 1923), on the reflection of electrons from metallic
surfaces. A complete verification of this radical hypothesis is furnished
by DYMOND'S experiments on the collisions of electrons in helium
(Nature, June 13, p. 910, 1925).

Unfortunately, the present state of quantum mechanics only
allows a qualitative description of these phenomena; for a complete
account of them the solution of the problem of the helium atom
would be necessary. It therefore seems particularly important to
explain the above-mentioned experiments of RUTHERFORD and his
co-workers on the dispersion of a-particles; for in this case we have
to do with a simple and completely known mechanism, the diffrac-
tion 5 of two charged particles by each other. The classical formula
which RUTHERFORD derived from a consideration of the hyperbolic
orbits of the particles, is experimentally verified for a large range;
but recently BLAGKETT has found departures from this law in the
encounters between a-particles and light atoms, and has suggested
that these might also be ascribed to diffraction effects of the DE
BROGLIE waves. At present only the preliminary question is settled,
of whether the classical formula can be derived as a limiting case
of quantum mechanics. G. WENTZEL (Zeit.f. Phys., Vol. 40, p. 590,
1926) has shown that this is in fact the case. The author of this
communication has, furthermore, carried through the computation
for the collision of electrons on the hydrogen atom, and arrived at
formulae which represent simultaneously the collisions of particles
of arbitrary energy (from slow electrons to fast a-particles). As yet
this has only been carried out for the first approximation, and so
gives no account of the more detailed diffraction effects. This
calculation thus yields a single expression for the Rutherford

PHYSICAL ASPECTS oi? QUANTUM MECHANICS 15

deflexion formula and the cross section of the hydrogen atom for
electrons in the range studied explicitly by LENARD. The same
method leads to a calculation of the probability of excitation of the
H-atom by electronic collision, but the calculations have not yet
been completed.

It would be decisive for the theory if it should prove possible to
carry the approximation further, and to see whether it fUrnishes an
explanation of the departures from the Rutherford formula.

Even, however, if these conceptions stand the experimental test,
it does not mean that they are in any sense final. Even now we can
say that they depend too much on the usual notion of space and
time. The formal quantum theory is much more flexible, and
susceptible of much more general interpretations. It is possible, for
example, to mix up co-ordinates and momenta by canonical
transformations, and so to arrive at formally quite different systems,
with quite different wave functions ijr. But the fundamental idea of
waves of probability will probably persist in one form or another.

ON THE SIGNIFICANCE OF COLLISION
PROCESSES IN THE UNDERSTANDING
OF QUANTUM MECHANICS

[First published in Proceedings of the International Conference of Physicists, Como 1927]

(QUANTUM mechanics, in its original matrix form due to
^^HEISENBERG, was suited only to the treatment of closed periodic
systems. It described possible states and transitions; it permitted
the calculation of the energy levels and of the oscillations of the
'virtual resonators' associated with the quantum jumps; but it
could not predict how a system would behave under given external
conditions. Soon, however, it was seen that, on the basis of matrix
mechanics, statistical statements at any rate are possible regarding
the behaviour of a system, provided that the latter is loosely con-
nected to another system. Its energy is then not constant, and the
matrix of the energy has non-diagonal elements, but the mean
value of the matrix is diagonal, and the element denoting the mean
energy in the ri- state under the action of the perturbation can be
regarded as the result of quantum jumps between the n^- state
and all the other states of the unperturbed system. To each jump
there belongs a transition probability which can be calculated
from the coupling. Nothing can be said, on the other hand,
regarding the moment when a quantum jump occurs. The further
development of quantum mechanics has made its statistical nature
more and more evident, especially when it became possible to treat
non-periodic processes. Of the extensions of the matrix calculus
which have been devised for this purpose, we mention the operator
calculus, which was introduced by the author together with N.
WIENER, DIRAG'S ^-number theory and the wave mechanics of
DE BROGUE and SGHRODINGER. The latter can be formally regarded
as a special case of the operator theory, although it grew from other
roots and brings to the fore important physical viewpoints; these
include the double nature of matter, which, like light, seems in
many ways to consist of waves and in other ways to consist of
corpuscles. The most general statement of the operator theory is as
follows. A physical quantity cannot in general be exactly specified
by giving the value of a co-ordinate; one can give only a frequency
law for its distribution over the whole range of variation of the
co-ordinate. Such a frequency law can in general be determined
only by an infinite number of numerical data, either by the variation

COLLISION PROCESSES IN QUANTUM MECHANICS 15

of a continuous function or by a sequence of discrete numbers;
these two modes of presentation are, however, not fundamentally
different, since, for example, a continuous function can be defined
by specifying the discrete sequence of its Fourier coefficients.
We therefore represent the distribution law in a wholly abstract
manner by a point in a space of infinitely many dimensions. A
Euclidean metric* can be introduced in this space; we then speak
of a Hilbert space. There are, however, not only rectangular sets
of discrete axes, but also sets of continuously distributed axes.
According to the kind of axis on which the point is projected, we
obtain one or the other of the two representations of the distribution
law, by number sequences or functions.

To every physical quantity there corresponds a linear operator,
i.e. an affine mapping of the Hilbert space on to itself, or, so to
speak, a homogeneous deformation of that space. Just as in the
theory of elasticity, there is always a system of principal axes dis-
tinguished by the fact that the points on the axes are only displaced
along the axes under the deformation. The magnitude of this
displacement, i.e. the values of the principal axes of the operator Q
considered, form the range of values which can be taken by the
physical quantity; this range may be continuous or discrete. The
position of the axes with respect to another system of axes is given
by an orthogonal matrix 0. An operator , whose principal axes
are known, can be associated with this other system of axes. The
elements of the matrix <j> are then functions of q' and ),', where
q' and Q are any two values (principal axes) of the two operatorsf
q and Q. This quantity $(q', Q} has, according to DIRAG and
JORDAN, a simple physical significance: | (j)(q'> Q) | 2 represents
the probability (or probability density) that, for a given Q, the
variable q' takes a given value (or lies in a given interval A#').
is called the probability amplitude. SCHRODINGER'S wave function
is a special case of this, namely the amplitude belonging to the
operators q and H(q, [A/27n']#/cty), where H(q,p) is HAMILTON'S
function; if we denote the principal axes of the latter, as is usual,
by W, then | <j>(q', W) [ 2 is the probability density that, for a
given energy, the co-ordinate q' lies in a given interval A#'.

We will not enter further into the elaboration of this formalism,
but ask instead what is the empirical evidence for this viewpoint.
This evidence consists, above all, of the atomic collision processes,
which almost compel us to interpret the square of the modulus of

* The expression for the distance, however, is not a quadratic but a Hermitian
form; all matrices representing physical quantities are not symmetric but
Hermitian.

f q' and Q,' niay span spaces of several dimensions.

jg COLLISION PROCESSES IN OJCJANTUM MECHANICS

SCHRODINGER'S wave function | <f>(q', W\Y as the number of
particles. For instance, if we take the case first investigated by
RUTHERFORD, where a beam of a particles collides with heavy
atomic nuclei, there corresponds to this a plane wave, which is
diffracted at the nucleus (by virtue of the Coulomb exchange inter-
action between the charges) and changed into a spherical wave.
WENTZEL and OPPENHEIMER have shown that one in fact obtains
RUTHERFORD'S formula for the number of scattered particles if the
intensity of the SchrSdinger wave is taken as a measure of the
probability. The probabilities of excitation and ionization can be
calculated, even for complex atoms, and one obtains the familiar
qualitative laws first discovered experimentally by FRANGK and
HERTZ, which form one of the most secure supports of the whole
quantum theory. ELSASSER has also investigated the retardation
of a particles by this method, and has shown that the well-known
classical theory of BOHR remains valid to some extent.

DIRAC has recently made a particularly important application
of this wave-mechanical collision theory by deriving the optical
dispersion formula with radiation damping. He regards the
process of scattering of light by atoms as a collision of the light
quanta with the atoms. Here it is sufficient to associate with the
atom two steady states: an upper in which the light quantum is
bound, and a lower in which it is free; in the latter case, the light
quantum has available a continuum of energy values. This simple
model suffices for the derivation of the dispersion formula, the
damping constant (line width) being expressed in terms of the
coupling between the atom and the light quantum. WIEN'S experi-
ments on the fading of the light emitted by canal rays can also be
interpreted in this way, and the same damping constant occurs.
A more exact investigation of the dependence of the damping
constant on the properties of the atom and of the spectral line
considered has yet to be made.

All these results confirm most impressively the statistical view of
quantum mechanics. The fundamental determinacy of natural
processes, always acknowledged in classical physics, must be
abandoned. The underlying reason for this lies in the dualism of
waves and corpuscles, which can be formulated as follows. To
describe natural processes, both continuous and discontinuous
elements are necessary. The appearance of the latter (corpuscles,
quantum jumps) is only statistically determined; the probability
of their appearance, however, is continuously propagated in the
manner of waves, which obey laws of a form similar to the causal
laws of classical physics.

ON THE MEANING OF PHYSICAL
THEORIES

[A lecture given at the public session of 10 November 1928. Nachrichten der
Gesellschaft der Wissenschaften zu G6ttingen, Geschaftliche Mitteilungen 1928-
29-]

IXTHOEVER regards in a detached way the development of the
* V exact sciences must be impressed by two contradictory features.
On the one hand, the whole of natural science exhibits a picture of
continuous and healthy growth, of unmistakable progress and
construction, evident as much in its inward deepening as in its
outward application to the technological mastery of Nature. Yet,
on the other hand, one observes at not infrequent intervals the
occurrence of upheavals in the basic concepts of physics, actual
revolutions in the world of ideas, whereby all our earlier knowledge
seems to be swept away, and a new epoch of investigation to be
inaugurated. The abrupt changes in the theories are in marked
contrast to the continuous flow and growth in the realm of well-
ascertained results. We may give a few examples of such convulsions
of theories. Consider the most ancient and most venerable branch
of physical science, astronomy, and the ideas concerning the stellar
universe, whose course we can follow through thousands of years.
At first, the Earth is at rest, a flat disc at the centre of the Universe,
round which the constellations move in orderly procession. Then,
almost simultaneously with the realization of the earth's size and
spherical shape, comes the Gopernican system of the Universe,
placing the sun in the centre and allotting to the earth only a
subordinate place among many other attendants of the central star.
The beginning of the new era in natural science is marked by New-
ton's theory of gravitation, which holds the solar system together,
and which remained unchallenged for some two centuries. In our
time, however, it has been dethroned by EINSTEIN'S relativistic
theory of gravitation, which completely does away both with the
heliocentric system of planets and with gravity acting at a distance.
The position is rather similar in optics, with its change in ideas
concerning the nature of light, imagined either as a stream of
small particles, according to NEWTON, or as a train of waves in
the light-ether, according to HUYGENS. At the beginning of the
nineteenth century there occurred the sudden change from the

1 8 ON THE MEANING OF PHYSICAL THEORIES

corpuscular theory to the wave theory; the present century, in turn,
brought with it a fresh transformation, of which I shall speak
presently. In the study of electricity and magnetism, the middle of
the last century was a time of revolution, in which the concept
of action at a distance was compelled to give place to the idea of a
continuous transmission of force through the ether. The profound
problem of the structure of matter, which chemistry a mighty
branch of the tree of physical sciences has made its especial con-
cern, exhibited even a few decades ago the immemorial antithesis
of atomistics and the continuum concept. This antithesis today
seems to be resolved in favour of the former; yet these problems are
bound up with one of the most fundamental revolutions of ideas,
which is taking place before our eyes under the name of the quantum
theory.

In a smaller scale the rise, acceptance and fall of theories is an
everyday occurrence; what today is valuable knowledge will
tomorrow be so much junk, hardly worth a historical backward
glance. The question thus arises : what then is the value of theories?
Are they not perhaps a mere by-product of research, a kind of
metaphysical ornament, draped like a lustrous cloak over the
'facts' which alone signify, at best a support and aid in our labour,
stimulants to the imagination in conceiving new experiments ?

The fact that this question can be proposed at all shows that the
meaning of physical theories is by no means obvious, and this is
why I have taken that subject as the theme of my lecture today.
There are many physicists at the present time, when once again a
grave crisis regarding the fundamental ideas of physics has just been
overcome, who are not entirely clear what to think of this latest
change of theory.

These theories relativity and the quantum theory which are
characteristic of the present time, are also the best suited for our
purposes, since we ourselves feel many of their assertions to be
strange, paradoxical, or even meaningless. The older theories must
have had a similar effect on their contemporaries; we, however,
can conjure up this state of mind only artificially, by historical
investigation. As I have paid little attention to the study of history,
I shall content myself with a brief glance backward to earlier periods
of crisis.

Any theoretical concept originates from observation and its
most plausible interpretation. The sight of the fixed, unshakable
earth on which we are borne, and of the moving heavens, leads
naturally to the geocentric system of the Universe. The fact that
light throws sharp shadows can be most simply understood in terms

ON THE MEANING OF PHYSICAL THEORIES 1 9

of the corpuscular hypothesis, which is found already, in poetical
form, in the works of LUCRETIUS. Of mechanics, which later
became a model for all physical theories, antiquity knew only
statics, the science of equilibrium. The reason is, of course, that the
forces acting upon levers and other machines can be replaced by
forces exerted by the human (or animal) body, and thus belong to
the realm of things directly perceptible to the senses.

What now is the significance of the change, when these primitive
ideas the geocentric system of the Universe, the corpuscular hypo-
thesis of light, the statical force in mechanics are replaced by
others ? The deciding factor is certainly Man's need to believe in a
real external world, independent of him and permanent, and his
ability to mistrust his sensations in order to maintain this belief.
A very distant object seems smaller than when it is near, but Man
sees always the 'object 5 , imagines it to be always the same size, and
believes with absolute certainty that he could go and convince
himself of the fact by touching and feeling the object. The objects
with which primitive Man deals stones, trees, hills, houses, animals,
men have the property of meeting this test. Such is the origin
of geometry, which in its beginnings was entirely the study of the
mutual positions and size relations of rigid bodies. In this sense
geometry is the most ancient branch of physics; it first showed
that objects in the external world. follow strict laws as regards their
spatial properties. Later, delight in the beauty of these laws had
the result that the empirical foundations of geometrical science
were disregarded or even denied, and the study of its logical frame-
work became an end in itself, as being a part of mathematics.
The geodesist and the astronomer, however, have always regarded
the teachings of geometry as statements concerning the real objects
in the world, and have never doubted that even bodies which,
because of their remoteness, are not directly accessible to us follow
the same laws. The application of the rules of geometry to the
planets showed that they must be very distant and very large, that
their motions on the night sky are only the projections of their true
paths in space; and finally the analysis of these paths and the
refinement of observational technique led of necessity to the COPER-
NIGAN system. The latter's victory proves that belief in well-tried
laws is stronger than a direct sense-impression. Of course, the new
theory must explain the reason for this sense-impression, on which
the previously accepted doctrine, now recognized to be false, was
based. In COPERNICUS' case, it sufficed to point out the size of
the earth in comparison with Man. This astronomical example
is typical of all subsequent cases. In the stellar universe we have

20 ON THE MEANING OF PHYSICAL THEORIES

for the first time a reality accessible to only one sense, that of
sight, and then often as an insignificant-seeming impression, far
removed from the lives and struggles of men, and yet undoubtedly
just as real as the chair in which I am sitting or the piece of paper
from which I am reading. This objective reality of which I speak
is always and everywhere founded on the same principle: obedience
to the general laws of geometry and physics. Even the chair I
regard as real only because it exhibits the constant properties
appertaining to solid bodies of its kind; the geometry and mech-
anics needed here is at everyone's command from unconscious
experience. There is no essential difference when we consider
the reasons why we think the point of light called Mars to be a
gigantic sphere like the earth; in this case, however, the obser-
vations must be more exact and geometry and mechanics must
be consciously applied. The simple and unscientific man's belief
in reality is fundamentally the same as that of the scientist. Some
philosophers concede this standpoint, as being practically indis-
pensable for the scientist; it goes under the name of empirical
realism and has a precarious position amongst the various kinds
of idealism. Here, however, we do not wish to discuss the quarrels
of the different schools of philosophy, but only to state as clearly
as possible the nature of the reality which forms the subject of
natural science. It is not the reality of sense-perceptions, of sensa-
tions, feelings, ideas, or in short the subjective and therefore absolute
reality of experience. It is the reality of things, of objects, which
form the substratum underlying perception. We take as a criterion
of this reality not any one sense-impression or isolated experience, but
only the accordance with general laws which we detect in phenomena.

What we have here expounded, using the example of astronomy,
occurs over and over again in the development of physics. We have
already ascertained in essentials the meaning of all theories, and
now wish to show that all the revolutions which have taken place in
physics are stages on the road to the construction of an objective
world, which combines the macrocosmos of the stars, the micro-
cosmos of the atoms and the cosmos of everyday things into a
consistent whole.

Let us first consider mechanics. In its period of simplicity it was,
as we have remarked, unable to progress beyond the study of
equilibrium. The study of motion or dynamics was the product of
a more sophisticated age. The laws which GALILEO and NEWTON
derived from their observations cannot be enunciated without ideas
which lie far outside the natural limits of thought. Words like
mass and force had, of course, been used earlier: 'mass 5 meant

ON THE MEANING OF PHYSICAL THEORIES 21

roughly the amount of some material, 'force' the magnitude of an
exertion. In mechanics, however, these words acquire a new precise
meaning; they are artificial words, perhaps the first to be coined.
Their sound is the same as words of ordinary speech, but their
meaning can be found only from a specially formulated definition.
I will not discuss this (by no means simple) definition here, but
merely mention that a concept occurs therein which, in the days
before science, played no part and can indeed be exactly explained
only with the aid of mathematical tools, namely the concept of
acceleration. If mass is defined by means of this concept (as 'resist-
ance to acceleration'), we already see clearly the foundation of
mechanics as an artificial product of the mind. Experience of
terrestrial bodies which could be adduced to support the new
theory in the period between GALILEO and NEWTON was fairly
limited. Yet the inner logic of Galilean mechanics was so strong
that NEWTON was able to take the great step of applying it to the
motions of the stars. The immense success of this step rests essenti-
ally on the idea that the force which the heavenly bodies exert on
one another is fundamentally the same as the gravity which we
know on earth. This idea, however, caused the abandonment of a
concept until then generally accepted, namely that forces from a
body are exerted only on its immediate neighbourhood. Only
such contact forces were known to statics. Terrestrial gravity, in
the work of GALILEO, at first appears only as a mathematical aid
in formulating the laws of falling. NEWTON himself regarded the
distant action from star to star, which he needed to explain the
motions of the planets, only as a provisional hypothesis, to be
later replaced by a contact or near-action. The effect of the
practical successes of NEWTON'S theory of gravitation on his succes-
sors was so overwhelming, however, that the distant action of gravi-
tation was not only taken for granted, but was used as a model for
the manner of action of other forces, those of electricity and
magnetism. Fierce battles have been fought in former times over
this distant action across empty space. Some called it a monstrosity
opposed to the natural idea of force; others hailed it as a marvellous
tool for unlocking the secrets of the stellar universe. Who was
right? We say: the Newtonian force of gravitation is an artificial
concept, which has little more than its name in common with the
simple idea, the feeling of force. Its justification rests merely on its
place in the system of objective natural science. So long as it fulfils
its duty there, it can remain; but as soon as new observations contra-
dict it, it must give way for the formation of new ideas, which will
be required to agree with the distant-action theory within the realm

22 ON THE MEANING OF PHYSICAL THEORIES

of the older observations. This change has occurred only in our
time, after a long development, which was closely connected with
the evolution of the sciences of electricity and magnetism.

As we have already said, the forces of electricity and magnetism
were, at the time of their first systematic investigation about 150
years ago, interpreted as distant actions on the model of gravi-
tation. COULOMB'S law of the attraction of electric charges, BIOT
and SAVART'S law of the effect of a current on a magnetic pole,
are imitations of NEWTON'S laws in form and conception. In the
mathematical construction of the theory, however, a notable thing
occurred: the so-called potential theory was found to give trans-
formations of these laws which put them in the form of near actions,
of forces exerted on one another by adjoining points in space.
Yet this remarkable equivalence of such heterogeneous concepts
went almost unnoticed. New discoveries had to be made in order
to compel a physical decision of the question 'distant or near
action ?' The discoverer of these new facts was FARADAY, and their
interpreter was MAXWELL. MAXWELL'S equations are a near-
action theory of electromagnetic phenomena, and thus signify
conceptually a return to a mode of thought closer to the natural
mode. I think, however, that this is quite unimportant. What
then is the state of affairs ? If we exclude FARADAY'S and MAXWELL'S
new discoveries, magnetic induction and dielectric displacement
current, MAXWELL'S equations contain nothing more than the
already existing potential theory, the mathematical transformation
of distant-action laws into near-action ones. The change in physical
theories occurring in the middle of the last cefatury is thus, from this
viewpoint, not really a revolution, destroying what exists, but a
conquest of new territory, involving a reorganization of the old
territory.

As a result of this conquest, however, a new concept comes to
the fore, that of the universal ether. For every near-action requires
a carrier, a substratum between whose particles the forces act, and
since the electric and magnetic forces can be transmitted even
through empty space, where no ordinary bodies are present, there
was nothing for it but to assume an artificial body. This, however,
was the easier inasmuch as such an ether had already been invented
in another field, that of optics, and the new theory of electricity
was in a position to identify this light-ether forthwith with the
electromagnetic ether.

We now come to the point where we can glance at the theory
of light. Here, as has already been remarked, the issue between the
corpuscular and wave theories had been decided in favour of the

ON THE MEANING OF PHYSICAL THEORIES 2$

latter at the beginning of the nineteenth century. Far-reaching as
this decision was, it signified, in the same sense as above, more a
conquest of new territory with consequent change of government
than a true revolution. For, so long as the phenomena of
interference and diffraction remained unknown, the concepts of
corpuscles and of very short waves were in actual fact equivalent,
so that the dispute could not be resolved. The fact that the whole
of the eighteenth century adhered to the corpuscular theory was
really an accident. Firstly, there was the authority of NEWTON,
who had preferred the corpuscular theory as being a simpler con-
cept, in the absence of cogent counter-proofs. Secondly, there
existed no mathematical proof that, even with short waves, the
occurrence of apparently sharp shadows can be explained; this
proof was first furnished by FRESNEL in trying to explain the actual
diffuseness of shadows, that is, the phenomena of diffraction. As
soon as these phenomena were discussed, there could no longer be
any doubt that the wave concept is the correct one. I should like
to emphasize that this is still true today, although, as we shall see
presently, the corpuscular theory has had a revival. Just as we
observe water waves and can follow their propagation, so we can
detect light waves with our apparatus. It would be entirely irrational
to employ different words and viewpoints in the two cases. This
certainty of the existence of light waves leads to the problematical
features of the most recent optical discoveries, which we shall
discuss below.

First of all, however, we must make a few remarks concerning
the ether problem. Waves require a carrier, and so it was assumed
that space is filled with the light-ether. The first period of the
ether theory again showed the simple carrying over of familiar
viewpoints. Elastic bodies were known to propagate waves, and so it
was assumed that the ether had the same properties as an ordinary
elastic substance. It could not, indeed, resemble a gas or a liquid,
since only longitudinal waves are propagated in the latter, whereas
experiments with polarized light show that light waves are certainly
transverse. It was thus necessary to assume a solid elastic ether
throughout the Universe, through which light waves are propagated.
It is obvious that this gives rise to difficulties when we try to under-
stand why the planets and the other heavenly bodies move through
this substance with no noticeable retardation. Nor was it possible
to explain satisfactorily the processes of reflection and refraction
at surfaces, propagation of light in crystals, and such like. It
was thus a relief when MAXWELL'S theory was experimentally
confirmed by HERTZ, since it was now possible to equate the

i>4 ON THE MEANING OP PHYSICAL THEORIES

electromagnetic ether with the light-ether. The formal difficulties
disappeared immediately, since the electromagnetic ether is not a
mechanical body with properties known from ordinary experience,
but an entity of a special kind, with its own laws like MAXWELL'S
equations, a typical artificial concept.

The period in physics following MAXWELL was so packed with
successes gained by this theory that the belief was often held that
all the essential laws of the inorganic world had been discovered.
For it proved possible to fit mechanics also into the 'electromagnetic
world picture', as it was called; the resistance to acceleration,
caused by the mass, was ascribed to electromagnetic induction
effects. Yet the limits of this realm were at hand, visible to the far-
seeing, and beyond those limits lay new territory which could not
be mastered by the means at hand. With this we enter the most
recent period. Its characteristic is that physical criticism takes in
ideas which no longer belong exclusively to its province, but are
claimed by philosophy as its own. Here, however, we shall always
place the physical viewpoints in the foreground.

As always, the conceptual difficulty came upon the theory of the
electromagnetic universal ether by a refinement in observational
technique: I refer to the celebrated Michelson-Morley experiment.
Before this, the ether could be imagined as a substance at rest
everywhere in the Universe, having particular properties, and
LORENTZ was able to show that all the then known electromagnetic
processes in bodies at rest or in motion could be explained in this
way. The real difficulty was to explain the fact that no ether wind
can be detected on the earth, which moves at a considerable speed
through the ether. LORENTZ was able to show that any optical and
electromagnetic effects caused by this ether wind must be extremely
small; they are proportional to the square of the ratio of the earth's
velocity to that of light, a quantity of the order of io~ 8 . Such small
quantities were below the limit of observability, until MIGHELSON'S
experiment was made. This should therefore have revealed the
blowing aside of light waves by the ether wind. It is well known,
however, that, like all later repetitions of the experiment, it showed
no trace of the effect. This was very difficult to explain, and very
artificial assumptions became necessary, such as the hypothesis put
forward by FITZ-GERAJLD and LORENTZ that all bodies are shortened
in the direction of their motion. The riddle was solved by EINSTEIN
in his 'special' theory of relativity, and the salient point in this was
a criticism of the idea of time.

What is time? To the physicist it is not the feeling of elapsion,
not the symbol of becoming and ceasing to be, but a measurable

ON THE MEANING OF PHYSICAL THEORIES 25

property of processes, like many others. In the naive period of
science, direct observation or perception of the passage of time
naturally determines the formation of the concept of time, and the
one-to-one correspondence between the passage of time and the
content of experience naturally led to the view that time is the
same here and everywhere else in the Universe. EINSTEIN was the
first to question whether this statement has any content that can
be tested empirically. He showed that the simultaneity of events
at different places can be ascertained only if an assumption is
made concerning the velocity of the signals used, and this, in
conjunction with the negative result of the ether-wind experiment,
led him to a new definition of simultaneity, which involved a
relativisation of the concept of time. Two events at different places
are not in themselves simultaneous; they may be so for one observer,
but not for another who is in motion relative to the first observer.
The physical concept of space was also caught up in this change in
ideas, especially when EINSTEIN, some years later, revealed the
relation of gravitation to the new conception of space and time. I
cannot enter into this 'general' theory of relativity within the limits
of this lecture; I will merely say that, in the theory of gravitation,
it signifies a transition from distant-action to near-action, and thus
an approach to intuitive ideas. On the other hand, it demands a
great step into the abstract: space and time lose all the simple
properties which before then had made geometry and motion
theory such convenient tools for physics. The familiar geometry of
EUCLID and the corresponding time are now reduced to mere
approximations to reality; but at the same time it becomes un-
intelligible why humanity has so far obtained such good results
with this approximation. Even today, one obtains satisfactory
results with it almost always in practice; in fact, it is an unfortunate
thing that the deviations capable of testing EINSTEIN'S theory are
very rare and difficult to observe. Together with the internal
consistency and logicality of the theory, however, they are enough
to gain it acceptance from physicists, apart from a few dissenters.
What is the position regarding the universal ether in the theory
of relativity? EINSTEIN at first proposed to avoid this concept
altogether. For the ether might be thought of as a substance having
at least the most elementary properties in common with ordinary
substances. These properties include the recognisability and identi-
fiability of individual particles. In the theory of relativity, however,
it is meaningless to say, C I have been at this point of the ether
before.' The ether would be a substance whose parts have neither
position nor velocity. Nevertheless, EINSTEIN later preferred to

2 6 ON THE MEANING OF PHYSICAL THEORIES

continue to use the word 'ether', as a purely artificial concept, of
course, having hardly anything in common with the ordinary idea
of a substance. For it is simply a grammatical necessity, in speaking
of oscillations and waves in space, to have a subject to govern the
verb 'to oscillate'. We therefore say, 'The ether oscillates, and does
so according to the field equations of EINSTEIN'S theory 5 ; and that
is all we can say about it.

The theory of relativity also modified importantly the concept of
mechanical mass, fusing it with that of energy. These are conse-
quences which are of the greatest significance in physics, in connec-
tion with investigations of the structure of matter and radiation;
they have not, however, aroused so much excitement as the criticism
of the traditional ideas of space and time, since the latter were
regarded as belonging to the content of philosophy. The fact of
the matter is as it is agreed by all sensible philosophers that
philosophy in former times, when the individual sciences had not
detached themselves, merely took over and retained the concep-
tions of natural science. Since these conceptions, as always in the
naive period, corresponded entirely to sense perception many
schools of thought formed the prejudice that they were an
immutable property of the mind, experience a priori. This is, of
course, true in the realm of perception, but not for the objective
realm of physics, whose properties must always be fitted to the
progress of experience and its systematic arrangement.

Much though the theory of relativity has brought in the way of
innovations, it is yet rather the climax of a development the
doctrine of the continuous universal ether than the inception of
a new period. A new period, however, does begin with the present
century by the introduction of PLANCK'S quantum theory. Its real
and deepest root is in atomistics, an ancient doctrine going back
to the Greek philosophers. Before 1900 it had developed quite
continuously and peacefully, though more and more richly and
fruitfully. Chemistry first made useful the concept of atoms;
gradually it conquered physics as well, mainly by explaining the
properties of gases and solutions, and from there penetrated into
the theory of electricity. The passage of electricity through electro-
lytic solutions -led to the hypothesis of atoms of electricity, called
electrons, and these were so brilliantly established in discharge
phenomena in gases, and in cathode rays and Becquerel rays, that
the reality of the electrons soon became as certain as that of the
material atoms. Now, when the electron had been revealed as a
kind of sub-atom, investigation was concentrated on the problem
of decomposing ordinary atoms into their electric component parts.

ON THE MEANING OF PHYSICAL THEORIES 27

The idea was that all atoms are built up of electrically negative
electrons and of electrically positive components whose nature was
not yet known. The difficulty is that, according to simple mathe-
matical theorems, charged bodies can never be at rest in stable
equilibrium under the known action of electric forces. It was thus
necessary to assume hypothetical unknown forces, and this is, of
course, rather unsatisfactory. Then came RUTHERFORD'S great
discovery. He bombarded atoms with atomic fragments, called
a-rays, emitted by radioactive bodies; these rays, by virtue of their
very high velocity, penetrate into the interior of the atoms they
strike. RUTHERFORD concluded with complete certainty from the
deflections undergone by the a-rays that they move as if a heavy
and very small positively charged mass, the 'nucleus', lay at the
centre of atom, this mass exerting the ordinary electric forces on
the a-particles. It thus became in the highest degree improbable
that the atom was held together by unknown non-electric forces.
But how could the electrons be in equilibrium around the nucleus ?
The only way out seemed to be to assume that the electrons are
not at rest, but move in orbits round the nucleus, like the planets
round the Sun. This, of course, did not help much, since such a
dynamical system is highly unstable. There is no doubt that
our planetary system would be reduced to chaos if it were so
unfortunate as to pass close to another large star; yet the atoms of
a gas survive a hundred million collisions every second, without
the slightest change in their properties.

This astonishing stability of atoms was an utter riddle from the
standpoint of the theory as it was at the end of the nineteenth
century, nowadays usually called the 'classical theory' for short.
An equally difficult puzzle was posed by the gigantic array of facts
which the spectroscopists had meanwhile assembled. Here one had
a direct message from the interior of the atom, in the form of light
oscillations emitted by it, and this message did not sound at all
like gibberish, but rather like an orderly language except that it
was unintelligible. For the gases, in particular, a simple structure
of the spectrum was recognisable: it consists of individual coloured
lines, each corresponding to a single periodic oscillation, and these
lines exhibited simple regularities. They can be arranged in series
in such a way that, from the serial number of the line, its position
in the spectrum can be calculated, with the greatest accuracy,
from a simple formula. This was first found by BALMER for hydrogen,
and later for many other substances by other investigators, in
particular RUNGE and RYDBERG. The attractive work of photo-
graphing and measuring spectra appealed to a great number of

28 ON THE MEANING OF PHYSICAL THEORIES

physicists, and so an immense quantity of observational material
was accumulated over the years, from which many important
conclusions could be drawn concerning individual problems in
physics, chemistry and astronomy, but whose real meaning remained
hidden. It was the same situation as with the extinct Maya peoples,
of whose script numerous specimens have been found in the ruined
cities of Yucatan; unfortunately, nobody can read them.

In physics, the key to the riddle was finally discovered, and that
by a strangely indirect road. At the turn of the century it was the
latest fashion to examine the radiation of glowing solid bodies.
Besides the technological importance of the problem in the manu-
facture of incandescent lamps and so on, profound theoretical
results were also hoped for from its solution. For KIRCHHOFF had
proved, on the basis of unassailable thermodynamic reasoning, that
radiation which leaves the interior of a glowing furnace through a
small hole must give a spectrum of an invariable kind, entirely
independent of the nature of the substances in the furnace and in
its walls; and this conclusion had been confirmed by experiment.
From the measurement of 'cavity radiation 5 , results were therefore
expected concerning quite general properties of the process of
radiation, and this expectation was not in vain. Nevertheless, it
now seems remarkable that one of the most profound laws could
be discovered in this way. For to resume the metaphor of a
foreign tongue one listened not to the articulate words of indi-
viduals, but to a crowd shouting all at once, and from this din the
key word was heard that made all the others intelligible. The
glowing furnace is such a complex structure, containing innumerable
oscillating atoms which send out to us their confused assembly of
waves. The characteristic feature of the spectrum of this assembly
is, by experiment, that it has a definite colour, according to the
temperature, red, yellow or white-hot. This means that a certain
range of oscillations, depending on the temperature, is most strongly
represented, while the intensity gradually falls to zero on both sides
of this, towards both rapid and slow oscillations. The classical
theory, on the other hand, demanded that the intensity should
continually increase on the side of rapid oscillations. Here there
was again an insoluble contradiction of the laws accepted at that
time.

After countless attempts to ascribe this contradiction to erroneous
conclusions within the classical theory had proved abortive, PLANCK
in 1900 ventured to propose a positive assertion amounting to this:
the energy of the oscillating particles in the furnace alters, not
continuously by radiation, but discontinuously, in jumps, and the

ON THE MEANING OF PHYSICAL THEORIES 29

ratio of the quantum of energy transferred in each jump to the
frequency of oscillations in the light emitted or absorbed is a fixed
and universal constant. This number, today known as PLANCK'S
constant, could be quite accurately calculated from experiments
then available on heat radiation, and has since been redetermined
many tunes by the most various methods, without any considerable
change in the original value.

In fact, a new fundamental constant of nature had been dis-
covered, comparable with the velocity of light or the charge on
the electron. This no one doubted, but most people found it very
difficult to accept the hypothesis of energy quanta. EINSTEIN alone
soon saw that it renders intelligible other peculiarities in the trans-
formation of mechanical energy into radiation. I must say a few
words regarding the most important of these phenomena, the so-
called photoelectric effect. If light of a given frequency falls on a
metal plate in a high vacuum, it is observed that electrons are
detached from the plate. The remarkable thing about the process
is that only the number, and not the velocity, of the electrons
emitted depends on the intensity of the light. The wave picture is
of no use in understanding this; for, if we move the metal plate
away from the light source, the incident wave becomes weaker and
more and more rarefied, and it is incomprehensible how it can
always communicate the same energy to an electron. EINSTEIN
observed that this behaviour can be immediately understood if the
light does not consist of waves, but is a shower of particles; the
hail of bullets from a machine-gun thins out with distance, but
each individual bullet retains its penetrating power. Combining
this idea with PLANCK'S quantum hypothesis, EINSTEIN predicted
that the energy of the light particle, and therefore that of the
ejected electron, must be equal to the frequency multiplied by
PLANCK'S constant. This result has been entirely confirmed by
experiments. Thus we have a revival of the old corpuscular theory
of light in a new form.

We shall consider below the conflict arising from this. First,
however, let me say a few words regarding the further development
of the quantum theory. It is well known that NIELS BOHR conceived
the idea of using PLANCK'S hypotheses to explain the properties of
atoms; he supposed that atoms (quite unlike a classical system of
planets) can exist only in a series of discrete states, and that, in a
transition from one state to another, light is emitted or absorbed
whose frequency is to the energy change of the atom in the ratio
given by PLANCK. By this means, all the contradictions mentioned
above between experiment and the classical theory are brought

go ON THE, MEANING OF PHYSICAL THEORIES

back to the same origin, and can be resolved by the assumption of
discrete energy quanta. The stability of the atom is explained by
the existence of a 'lowest' quantum state, in which the atom remains
even when perturbed, provided that the perturbations do not reach
the amount of the smallest energy jump possible in the atom. The
existence of this lowest energy threshhold was established experi-
mentally by FRANCK and HERTZ, who bombarded atoms (of mercury
vapour) with electrons of measured velocity. At the same time,
this confirmed BOHR'S hypothesis concerning light emission; for as
soon as the energy of the bombarding electrons exceeded the first
energy threshhold, light of a single colour was emitted, its frequency
being that calculated from the energy by means of PLANCK'S
relation. The whole of the large amount of observational material
accumulated by the spectroscopists was thus converted, at one
stroke, from a collection of numbers and unintelligible rules to a
most invaluable record regarding the possible states of the atoms
and the energy differences between them. Further, the previously
quite enigmatic conditions for the excitation of the various spectra
became completely intelligible.

Despite this enormous success of BOHR'S point of view, the road
from his simple idea of stationary states to a complete and logically
satisfactory mechanics of the atom was a long and laborious one.
Here again we have the primitive period, in which the laws of
ordinary mechanics were applied as far as possible to the electron
orbits in the atom, and it is remarkable that this was in fact possible
to some extent, despite the irreconcilable antithesis between the
continuous nature of the classical quantities and the discontinuous
processes (jumps) of the quantum theory. Finally, however, the
necessary modification of mechanics was effected, so as to take
account of the discontinuities. The new quantum mechanics was
evolved in different forms, partly from a fundamental idea due to
HEISENBERG, by one group, here in Gottingen and by DIRAC in
Cambridge, partly as the so-called wave mechanics of DE BROGLIE
and SCHRODINGER. These formalisms finally proved to be essentially
identical; together, they form a logically closed system, the equal
of classical mechanics in internal completeness and external
applicability. At first, however, they were only formalisms, and it
was a matter of discovering their meaning a posteriori. It is, in fact,
very common in physical investigations to find it easier to derive
a formal relation from extensive observational material than to
understand its real significance. The reason for this lies deep in the
nature of physical experience: the world of physical objects lies
outside the realm of the senses and of observation, which only

ON THE MEANING OF PHYSICAL THEORIES 31

border on it; and it is difficult to illuminate the interior of an
extensive region from its boundaries. In the quantum theory,
there were especial difficulties, of which I should like to discuss the
most important, namely the revival of the corpuscular theory of
light. The idea of individually moving light quanta was supported
by a number of further tests, and in particular by GOMPTON'S
experiment. This showed that, when such a light quantum collides
with an electron (realised as the scattering of X-rays by substances,
such as paraffin, with many loosely-bound electrons) the usual
collision laws of mechanics hold, as for billiard balls. The primary
light quantum gives up some energy to the electron with which it
collides, and so the recoiling light quantum has less energy, and
by PLANCK'S relation a smaller frequency than the primary one.
The consequent decrease in frequency of the scattered X-ray has
been demonstrated experimentally, and so has the existence of
recoil electrons.

There is thus no doubt of the correctness of the assertion that
light consists of particles. But the other assertion that light consists
of waves is just as correct. In discussing the proofs of the wave
nature of light, we have seen that, in every phenomenon of inter-
ference, we can perceive the light waves as clearly and evidently as
water waves or sound waves. The simultaneous existence of
corpuscles and waves, however, seems quite irreconcilable. Never-
theless, the theory must solve the problem of reconciling these two
ideas, not of course in the realm of observation, but in that of
objective physical relations, where the only criterion of existence,
apart from freedom from logical contradiction, is agreement between
theoretical -predictions and experiment. The solution of this
problem was attained by a criticism of fundamental concepts,
very similar to that in the theory of relativity.

The basis of the entire quantum theory is PLANCK'S relation
between energy and frequency, which are asserted to be propor-
tional. In this 'quantum postulate', however, there is an absurdity.
For the concept of energy clearly refers to a single particle (a light
quantum or an electron), that is, to something of small extent; the
concept of frequency, however, belongs to a wave, which must
necessarily occupy a large region of space, and indeed, strictly
speaking, the whole of space: if a segment of a purely periodic
wave-train is removed, it is no longer periodic. The equating of
the energy of a particle and the frequency of a wave is thus in
itself quite irrational. It can, however, be made rational, if a
principle is renounced which was previously always accepted in
physics, namely, that of determinism. Earlier, it had always been

32 OK THE MEANING OF PHYSICAL THEORIES

supposed that the photoelectric process, in which an electron is
ejected from a metal plate by a light wave, is determined in every
detail that there is meaning in the question, 'When and where is
an electron ejected? 3 Or, what is the same thing, "Which light
quantum, at what point and at what time, takes effect on striking
the plate? 5

Suppose that we decide to renounce this question, an act which
is the easier inasmuch as no experimenter would think of asking it,
or answering it, in a particular case. It is, in fact, clearly a purely
artificial question; the experimenter is invariably content to find
out how many particles appear, and with what energy.

Let us therefore not ask where exactly a particle is, but be satisfied
to know that it is in a definite, though fairly large, region of space.
The contradiction between the wave and corpuscular theories then
disappears. This is most easily seen if we allot to the wave the
function of determining the probability that a particle will appear,
the energy of the particle being related to the frequencies present
in the wave by means of PLANCK'S relation. If the region of space
considered is large, and the wave-train consequently almost un-
perturbed and purely periodic, there corresponds to it a precise
frequency, and a precisely defined particle energy; but the point
where the particles appear in this region of space is quite indefinite.
If it is desired to determine the position of the particles more
exactly, the region of space in which the process is observed must
be diminished; by so doing, however, a segment of the wave is
removed, and its purely periodic character is destroyed; such
a non-periodic disturbance, nevertheless, can be analysed into a
greater or lesser number of purely periodic oscillations; to each of
the various frequencies of this mixture, there then corresponds a
different energy of the observed particles. Thus an exact deter-
mination of position destroys the determination of the energy, and
vice versa.

This law of restricted measurability discovered by HEISENBERG
has been confirmed in every case. For every extensive quantity
(such as determinations of position and time), there is an intensive
quantity (such as velocity and energy), such that, the more exactly
the one is determined, the less accurately can the other be deter-
mined, and it is found that the product of the ranges to within
which two such associated quantities are known is exactly PLANCK'S
constant. That is the true significance of this hitherto mysterious
constant of nature; it is the absolute limit of accuracy of all measure-
ments. Only its extreme smallness is responsible for the fact that
its existence was not discovered earlier.

ON THE MEANING OF PHYSICAL THEORIES 3

From this standpoint it is possible to interpret the formalism of
quantum mechanics, in any individual case, so that the relation
with the observational concepts of the experimenter is shown,
without the possibility of any contradiction.

This, of course, does not happen without the sacrifice of familiar
ideas. For example, when we speak of a particle, we are accustomed
to imagine its entire path in a concrete manner. We may continue
to do so, but we must be careful in drawing conclusions therefrom.
For, if such an assumed path is to be tested experimentally, the test
itself will in general alter the path, no matter how carefully it is
performed. More fundamentally important is the abandonment of
determinism, the replacement of a rigorously causal description
by a statistical one.

Probability and statistics have already played a certain part in
physics, in the case of phenomena involving large numbers (e.g. in
the kinetic theory of gases). These methods, however, were usually
regarded as emergency devices in cases where our knowledge of
details is insufficient. Provided that the position and velocity of
all the particles in a closed system were known at some instant,
the future evolution of the system would be completely determined,
and could be predicted by mere calculation. This corresponds to
our experience concerning large bodies. Let us recall the story of
William Tell. When Tell, before aiming at the apple, sent a
brief prayer to Heaven, he surely prayed for a steady hand and a
keen eye, believing that the arrow would then find its way into the
apple automatically. In precisely the same way, the physicist
supposed that his electron and a-ray bullets would certainly hit
any desired atom, provided that he could aim accurately enough,
and he did not doubt that this was merely a question of practice,
which could be solved better and better as experimental technique
progressed. Now, on the contrary, it is asserted that the aiming itself
can be only of limited accuracy. If Gessler had ordered Tell to
shoot a hydrogen atom from his son's head by means of an a-
particle, and given him, instead of a crossbow, the best laboratory
instruments in the world, then TelTs skill would have been un-
availing; whether he hit or missed would have been a matter of
chance.

The impossibility of exactly measuring all the data of a state
prevents the predetermination of the further evolution of the system.
The causality principle, in its usual formulation, thus becomes
devoid of meaning. For if it is in principle impossible to know all
the conditions (causes) of a process, it is empty talk to say that every
event has a cause. Of course, this opinion will be opposed by those

34 ON THE MEANING OF PHYSICAL THEORIES

who see in determinism an essential feature of natural science
There are others, however, who hold the contrary opinion that
quantum mechanics asserts nothing new as regards the question
of determinism; that, even in classical mechanics, determinism is
only a fiction and of no practical significance*; that, in reality,
despite mechanics, there holds everywhere the principle that the
basis of all statistics is small causes, great effects. If, for instance,
we consider the atoms of a gas as small spheres, the mean free path
between two collisions is, at normal pressure, many thousands of
times the diameter of the atom; a very slight deviation in the
direction of recoil at one collision will therefore convert a direct
hit at the next collision into a miss, and a marked change of direction
will be replaced by an undisturbed passage. This is certainly so,
but it does not yet reach the heart of the matter. Let us return once
more to Tell What better example could we have of the theorem
of a small cause and a great effect than shooting at the apple, where
the accuracy of the aim is a matter of life and death? Yet the story
is evidently based on the conception of the ideal marksman, who
can always make the error of his aim smaller than the most diminu-
tive target supposing, of course, that no unforeseeable influence,
such as the wind, diverts his missile. In exactly the same way, we
can imagine an ideal case in classical mechanics; a system com-
pletely isolated from external influences and an exactly determined
initial state, and there is no reason to suppose that any approxima-
tion to this aim is not only difficult but impossible. Quantum
mechanics, however, asserts that it may be impossible. This
distinction may seem pointless to the practical scientist; the
discovery of the existence of an absolute limit of accuracy is, however,
of great importance in the logical structure of the theory.

Even if we disregard all philosophical aspects, the contradiction
between the corpuscular and wave properties of radiation would be
insoluble in physics without this statistical viewpoint. This is
where the theory has scored a great success : it predicted on formal
grounds that even material rays, of emitted atoms or electrons,
must exhibit a wave character in suitable circumstances, and the
experimenters have since confirmed this prediction by remarkable
interference experiments.

Although the new theory then seems well founded on experiment,
it may still be asked whether it cannot in future be made again
deterministic by extension or refinement. To this we may reply

* R. VON MISES, Probability, Statistics and Truth, Springer, 1928. Compare
the Mowing argument with the article "Is Classical Mechanics in fact Deter-
ministic? ", p. 164 of this collection.

ON THE MEANING OF PHYSICAL THEORIES 35

that it can be proved by exact mathematics that the accepted
formalism of quantum mechanics admits of no such addition. If
therefore it is desired to retain the hope that determination will
some day return, the present theory must be regarded as intrinsically
false; certain statements of this theory would have to be disproved
by experiment. The determinist must therefore not protest but
experiment, if he wishes to convert the adherents of the statistical
theory.

Of course many people 3 on the contrary, welcome the abandon-
ment of determinism in physics. I remember that, at the time of
the appearance of the earliest work on the statistical interpretation
of quantum mechanics, a gentleman approached me with some
occultistic pamphlets, thinking I might be suitable for a conversion
to spiritualism. There are also, however, serious observers of scientific
evolution, who consider the present turn in physics to be the collapse
of one conception of the Universe and the beginning of another,
deeper idea of the nature of 'reality'. Physics itself, they claim,
admits that there are 'gaps in the sequence of determinateness'.
What right has it then to put forward its devices as 'realities' ?

In meeting such arguments it is important to demonstrate clearly
that the new quantum mechanics is no more and no less revolu-
tionary than any other newly propounded theory. Once again, it
is really a conquest of new territory; in the course of this it is
found, as on previous occasions, that the old principles are no longer
wholly adequate, and must be in part replaced by new ones. But
the old ideas still remain as a limiting case, comprising all phenomena
for which PLANCK'S constant can, on account of its smallness, be
neglected in comparison with quantities of the same kind. Thus
events in the world of large bodies obey to a high accuracy the old
deterministic laws; deviations occur only in the atomic range. If
quantum mechanics has any peculiarity, it is that it does not decide
between two modes of presentation (corpuscles and waves) which
previously were equally possible, but, after the seeming victory of
one, reinstates the other and combines both in a higher unity.
The necessary sacrifice is the idea of determinism; but this does
not mean that rigorous laws of Nature no longer exist. Only the
fact that determinism is among the ordinary concepts of philosophy
has caused us to regard the new theory as particularly revolu-
tionary.

I hope to have shown that the whole evolution of physical
theories, up to their latest form, is governed by a consistent striving,
and the object of this striving will be clear from the individual
examples given. Let me attempt to express it once more in a

36 ON THE MEANING OF PHYSICAL THEORIES

somewhat more general form. The world of Man's experience is
infinitely rich and manifold, but chaotic and involved with the
experiencing being. This being strives to arrange his impressions
and to agree with others concerning them. Language, and art
with its numerous modes of expression, are such ways of transmission
from mind to mind, complete in their way where objects of the sense-
world are concerned, but not well suited to the communication of
exact ideas concerning the outer world. This marks the beginning
of the task of science. From the multitude of experiences it selects a
few simple forms, and constructs from them, by thought, an objective
world of things. In physics, all 'experience' consists of the activity
of constructing apparatus and of reading pointer instruments. Yet
the results thereby obtained suffice to re-create the cosmos by
thought. At first images are formed which are much influenced
by observation; gradually, the conceptions become more and
more abstract; old ideas are rejected and replaced by new ones.
But, however far the constructed world of things departs from
observation, nevertheless it is indissolubly linked at its boundaries
to the perceptions of the senses, and there is no statement of the
most abstract theory that does not express, ultimately, a relation
between observations. That is why each new observation shakes
up the entire structure, so that theories seem to rise and fall. This,
however, is precisely what charms and attracts the scientist. The
creation of his mind would be a melancholy thing, did it not die
and come to life once more.

SOME PHILOSOPHICAL ASPECTS OF
MODERN PHYSICS

[Inaugural Lecture as Tait Professor of Natural Philosophy, University of
Edinburgh. First published in Proc. Roy. Soc. Edinburgh, Vol. LVTI, Part I, pp.
1-18, 1936-37.]

HpHE Chair which I have been elected to occupy, in succession
A to Professor DARWIN, is associated with the name of a great
scholar of our fathers' generation, PETER GUTHRIE TAIT. This
name has been familiar to me from the time when I first began to
study mathematical physics. At that time FELIX KLEIN was the
leading figure in a group of outstanding mathematicians at Got-
tingen, amongst them HILBERT and MDSTKOWKSI. I remember how
KLEIN, ever eager to link physics with mathematics, missed no
opportunity of pointing out to us students the importance of
studying carefully the celebrated Treatise on Natural Philosophy of
THOMSON and TAIT, which became a sort of Bible of mathematical
science for us.

To-day theoretical physics has advanced in very different direc-
tions, and 'Thomson and Tait' is perhaps almost unknown to the
younger generation. But such is the fate of all scientific achievement;
for it cannot claim eternal validity like the products of great artists,
but has served well if it has served its time. For myself this book has
a special attraction by reason of its title. The subject known every-
where else in the world by the dull name 'Physics' appears here
under the noble title of 'Natural Philosophy, 'the same title as is
given to the two Chairs of Physics in this University. Our science
acquires by virtue of this name a dignity of its own. Occupied by
his tedious work of routine measurement and calculation, the
physicist remembers that all this is done for a higher task: the
foundation of a philosophy of nature. I have always tried to think
of my own work as a modest contribution to this task; and in
entering on the tenure of the Tait Chair of Natural Philosophy at
this University, though far from my fatherland, I feel intellectually
at home.

The justification for considering this special branch of science
as a philosophical doctrine is not so much its immense object, the
universe from the atom to the cosmic spheres, as the fact that the
study of this object in its totality is confronted at every step by

37

38 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

logical and epistemological difficulties; and although the material
of the physical sciences is only a restricted section of knowledge,
neglecting the phenomena of life and consciousness, the solution
of these logical and epistemological problems is an urgent need
of reason.

For describing the historical development it is a convenient
coincidence that the beginning of the new century marks the separa-
tion of two distinct periods, of the older physics which we usually
call classical, and modern physics. EINSTEIN'S theory of relativity
of 1905 can be considered as being at once the culmination of
classical ideas and the starting-point of the new ones. But during
the preceding decade research on radiation and atoms, associated
with the names of RONTGEN, J. J. THOMSON, BECQUEREL, the
CURIES, RUTHERFORD, and many others, had accumulated a great
number of new facts which did not fit into the classical ideas at
all. The new conception of the quantum of action which helped
to elucidate them was first put forward by PLANCK in 1900. The
most important consequences of this conception were deduced by
EINSTEIN, who laid the foundations of the quantum theory of light
in 1905, the year in which he published his relativity theory, and
by NIELS BOHR in 1913, when he applied the idea of the quantum
to the structure of atoms.

Every scientific period is in interaction with the philosophical
systems of its time, providing them with facts of observation and
receiving from them methods of thinking. The philosophy of the
nineteenth century on which classical physics relied is deeply rooted
in the ideas of DAVID HUME. From his philosophy there developed
the two systems which dominated science during the latter part of
the classical period, critical philosophy and empiricism.

The difference between these systems concerns the problem of
the a priori. The idea that a science can be logically reduced to a
small number of postulates or axioms is due to the great Greek
mathematicians, who first tried to formulate the axioms of geometry
and to derive the complete system of theorems from them. Since
then the question of what are the reasons for accepting just these
axioms has perpetually occupied the interest of mathematicians and
philosophers. KANT'S work can be considered as a kind of enormous
generalization of this question; he attempted to formulate the
postulates, which he called categories a priori, necessary to build up
experience in general, and he discussed the roots of their validity.
The result was the classification of the a priori principles into two
classes, which he called analytic and synthetic, the former being
the rules of pure logical thinking, including arithmetic, the latter

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 39

containing the laws of space and time, of substance, causality, and
other general conceptions of this kind. KANT believed that the root
of the validity of the first kind was c pure reason' itself, whereas the
second kind came from a special ability of our brain, differing from
reason, which he called 'pure intuition' (reine Anschauung) . So mathe-
matics was classified as a science founded on a priori principles, pro-
perties of our brain and therefore unchangeable; and the same was
assumed for some of the most general laws of physics, as formulated
by NEWTON.

But I doubt whether KANT would have maintained this view if he
had lived a little longer. The discovery of non-Euclidean geometry
by LOBATGHEFSKY and BOLYAI shook the a priori standpoint. GAUSS
has frankly expressed his opinion that the axioms of geometry have
no superior position as compared with the laws of physics, both
being formulations of experience, the former stating the general
rules of the mobility of rigid bodies and giving the conditions for
measurements in space. Gradually most of the physicists have been
converted to the empirical standpoint. This standpoint denies the
existence of a priori principles in the shape of laws of pure reason
and pure intuition; and it declares that the validity of every state-
ment of science (including geometry as applied to nature) is based
on experience. It is necessary to be very careful in this formulation.
For it is, of course, not meant that every fundamental statement
as, for instance, the Euclidean axioms of geometry is directly based
on special observations. Only the totality of a logically coherent
field of knowledge is the object of empirical examination, and if a
sufficient set of statements is confirmed by experiment, we can
consider this as a confirmation of the whole system, including the
axioms which are the shortest logical expression of the system.

I do not think that there is any objection to this form of empiric-
ism. It has the virtue of being free from the petrifying tendency
which systems of a priori philosophy have. It gives the necessary
freedom to research, and as a matter of fact modern physics has
made ample use of this freedom. It has not only doubted the a
priori validity of Euclidean geometry as the great mathematicians
did a hundred years ago, but has really replaced it by new forms of
geometry; it has even made geometry depend on physical forces,
gravitation, and it has revolutionized in the same way nearly all
categories a priori, concerning time, substance, and causality.

This liberation from the idea of the a priori was certainly important
for the development of science, but it already took place during the
last century, and does not represent the deciding difference between
classical and modern physics. This difference lies in the attitude to

40 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

the objective world. Classical physics took it for granted that there
is such an objective world, which not only exists independently of
any observer, but can also be studied by this observer without
disturbing it. Of course every measurement is a disturbance of the
phenomenon observed; but it was assumed that by skilful arrange-
ment this disturbance can be reduced to a negligible amount.
It is this assumption which modern physics has shown to be wrong.
The philosophical problem connected with it arises from the diffi-
culty in speaking of the state of an objective world if this state
depends on what the observer does. It leads to a critical examination
of what we mean by the expression 'objective world.'

The fact that statements of observations depend on the stand-
point of the observer is as old as science. The orbit of the earth
round the sun is an ellipse only for an observer standing just at the
centre of mass of the two bodies. Relativity gave the first example
in which the intrusion of the observer into the description of facts
is not so simple, and leads to a new conception to conserve the
idea of an objective world. EINSTEIN has acknowledged that his
studies on this problem were deeply influenced by the ideas of
ERNST MACH, a Viennese physicist who developed more and more
into a philosopher. From his writings sprang a new philosophical
system, logical positivism, which is much in favour to-day. Traces of
it can be seen in fundamental papers of HEISENBERG on quantum
theory; but it has also met with strenuous opposition, for instance,
from PLANCK. In any case, positivism is a living force in science.
It is also the only modern system of philosophy which by its own
rules is bound to keep pace with the progress of science. We are
obliged to define our attitude towards it.

The characteristic feature of this system is the sharp distinction
it draws between real and apparent problems, and correspondingly
between those conceptions which have a real meaning and those
which have not. Now it is evident and trivial that not every gram-
matically correct question is reasonable; take, for instance, the
well-known conundrum: Given the length, beam, and horse-power
of a steamer, how old is the captain? or the remark of a listener
to a popular astronomical lecture: C I think I grasp everything, how
to measure the distances of the stars and so on, but how did they
find out that the name of this star is Sirius?' Primitive people are
convinced that knowing the 'correct 5 name of a thing is real know-
ledge, giving mystical power over it, and there are many instances
of the survival of such word-fetishism in our modern world. But
let us now take an example from physics in which the thing is not
so obvious. Everybody believes he knows what the expression

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 41

'simultaneous events' means, and he supposes as a matter of
course that it means the same for any other individual. This is
quite in order for neighbours on this little planet. Even when
science made the step of imagining an individual of similar brain-
power on another star there seemed to be nothing problematical.
The problem appeared only when the imagination was driven so
far as to ask how an observer on the earth and another on, say,
Mars could compare their observations about simultaneous events.
It was then necessary to take into account the fact that we are
compelled to use signals for this comparison. The fastest signal at
our disposal is a flash of light. In using light, or even only thinking
about it, we are no longer permitted to rely on our brainpower, our
intuition. We have to consider facts revealed by experiments. We
have not only the fact of the finite velocity of light, but another
most important fact, disclosed by MICHELSON'S celebrated experi-
ment: that light on this earth travels with the same speed in all
directions, independently of the motion of the earth round the sun.
One usually expresses this by saying that these experiments disprove
the existence of an ether-wind which we would expect from the
analogy of the wind felt in a moving car.

An admirable logical analysis of these facts led EINSTEIN to the
result that the question of simultaneity of two distant events is
almost as absurd as that regarding the age of the captain. Just as
this question would become significant by adding some data, say
about his life insurance, the problem of simultaneity becomes
reasonable by adding data about the motion of the observer. In
this way the conception of time loses its absolute character, and
space becomes involved in this revolution. For it becomes meaning-
less to speak about 'space at this moment' ; if we assume two ob-
servers in relative motion just passing one another, then each has
his own 'space at this moment,' but the events contained in this
space are different for the two observers.

What has now become of the idea of a world independent of the
observer? If one sticks to the meaning of a static assembly of things
at one moment, this idea of an objective world is lost. But it can be
saved by considering as the world the assembly of events, each
having not only a given position in space but also a given time of
occurrence. MINKOWSKI has shown that it is possible to get a
description of the connection of all events which is independent of
the observer, or invariant, as the mathematicians say, by con-
sidering them as points in a four-dimensional continuum with a
quasi-Euclidean geometry. But the division of this four-dimensional
world into space and time depends on the observer.

42 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

When I wrote a popular book on relativity in 1920 I was so
impressed by this wonderful construction that I represented this
method of objectivation as the central achievement of science.
I did not then realise that we were soon to be confronted with a
new empirical situation which would compel us to undertake a
much deeper critical review of the conception of an objective
world.

I have here used the phrase 'new empirical situation, 5 following
NLELS BOHR, the founder of modern atomic theory, and the deepest
thinker in physical science. He has coined this expression to indicate
that the birth of new and strange ideas in physics is not the result
of free or even frivolous speculation, but of the critical analysis of
an enormous and complicated body of collected experience.
Physicists are not revolutionaries but rather conservative, and
inclined to yield only to strong evidence before sacrificing an
established idea. In the case of relativity this evidence was strong
indeed, but consisted to a large extent of negative statements,
such as that mentioned above regarding the absence of an ether-
wind. The generalization which was conceived by Einstein in 1915
combining the geometry of the space-time world with gravitation
rested, and still rests, on a rather slender empirical basis.

The second revolution of physics, called quantum theory, is,
however, built on an enormous accumulation of experience, which
is still growing from day to day. It is much more difficult to talk
about these matters, because they have a much more technical
character. The problem is the constitution of matter and radiation,
which can be adequately treated only in laboratories with refined
instruments. The evidence provided there consists of photographic
plates, and of tables and curves representing measurements. They
are collected in enormous numbers all over the world, but known
only to the experts. I cannot suppose that you are acquainted with
these experiments. In spite of this difficulty, I shall try to outline
the problem and its solution, called quantum mechanics.

Let us start with the old problem of the constitution of light. At
the beginning of the scientific epoch two rival theories were pro-
posed: the corpuscular theory by NEWTONJ the wave theory by
HUYGENS. About a hundred years elapsed before experiments were
found deciding in favour of one of them, the wave theory, by the
discovery of interference. When two trains of waves are superposed,
and a crest of one wave coincides with a valley of the other, they
annihilate one another; this effect creates the well-known patterns
which you can observe on any pond on which swimming ducks or
gulls excite water-waves. Exactly the same kind of pattern can be

SOME PHILOSOPHICAL ASPECTS OP MODERN PHYSICS 43

observed when two beams of light cross one another, the only
difference being that you need a magnifying-lens to see them;
the inference is that a beam of light is a train of waves of short
wave-length. This conclusion has been supported by innumerable
experiments.

But about a hundred years later, during my student days,
another set of observations began to indicate with equal cogency
that light consists of corpuscles. This type of evidence can best
be explained by analogy with two types of instruments of war,
mines and guns. When a mine explodes you will be killed if you
are near it, by the energy transferred to you as a wave of compressed
air. But if you are some hundred yards away you are absolutely
safe; the explosion-wave has lost its dangerous energy by con-
tinuously spreading out over a large area. Now imagine that the
same amount of explosive is used as the propellant in a machine-
gun which is rapidly fired, turning round in all directions. If you
are near it you will almost certainly be shot, unless you hastily run
away. When you have reached a distance of some hundred yards
you will feel much safer, but certainly not quite safe. The pro-
bability of being hit has dropped enormously, but if you are hit
the effect is just as fatal as before.

Here you have the difference between energy spread out from a
centre in the form of a continuous wave-motion, and a discon-
tinuous rain of particles. PLANCK discovered, in 1900, the first
indication of this discontinuity of light in the laws governing the
heat radiated from hot bodies. In his celebrated paper of 1905,
mentioned already, EINSTEIN pointed out that experiments on the
energetic effect of light, the so-called photoelectric effect, could be
interpreted in the way indicated as showing unambiguously the
corpuscular constitution of light. These corpuscles are called quanta
of light or photons.

This dual aspect of the luminous phenomenon has been confirmed
by many observations of various types. The most important step
was made by BOHR, who showed that the enormous amount of
observations on spectra collected by the experimentalists could
be interpreted and understood with the help of the conception of
light-quanta. For this purpose he had also to apply the idea of
discontinuous behaviour to the motion of material particles, the
atoms, which are the source of light.

I cannot , follow out here the historical development of the
quantum idea which led step by step to the recognition that we
have here to do with a much more general conception. Light is
not the only 'radiation' we know; I may remind you of the cathode

44 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

rays which appear when electric currents pass through evacuated
bulbs, or the rays emitted by radium and other radioactive sub-
stances. These rays are certainly not light. They are beams of
fast-moving electrons, i.e. atoms of electricity, or ordinary atoms
of matter like helium. In the latter case this has been proved
directly by RUTHERFORD, who caught the beam (a so-called a-ray
of radium) in an evacuated glass vessel and showed that it was
finally filled with helium gas. To-day one can actually photograph
the tracks of these particles of radiating matter in their passage
through other substances.

In this case the corpuscular evidence was primary. But in 1924
DE BROGLIE, from theoretical reasoning, suggested the idea that
these radiations should show interference and behave like waves
under proper conditions. This idea was actually confirmed by
experiments a short time later. Not only electrons, but real atoms
of ordinary matter like hydrogen or helium have all the properties
of waves if brought into the form of rays by giving them a rapid
motion.

This is a most exciting result, revolutionising all our ideas of
matter and motion. But when it became known, theoretical physics
was already prepared to treat it by proper mathematical methods,
the so-called quantum mechanics, initiated by HEISENBERG, worked
out in collaboration with JORDAN and myself, and quite inde-
pendently by DIRAC; and another form of the same theory, the
wave-mechanics, worked out by SCHRODINGER in close connection
with DE BROGUE'S suggestion. The mathematical formalism is a
wonderful invention for describing complicated things. But it does
not help much towards a real understanding. It took several years
before this understanding was reached, even to a limited extent,
But it leads right amidst philosophy, and this is the point about
which I have to speak.

The difficulty arises if we consider the fundamental discrepancy
in describing one and the same process sometimes as a rain of
particles, and at other times as a wave. One is bound to ask, what
is it really? You see here the question of reality appears. The
reason why it appears is that we are talking about particles or
waves, things considered as well known; but which expression is
adequate depends on the method of observation. We thus meet a
situation similar to that in relativity, but much more complicated.
For here the two representations of the same phenomenon are
not only different but contradictory. I think everyone feels that a
wave and a particle are two types of motion which cannot easily
be reconciled. But if we take into account the simple quantitative

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 45

law relating energy and frequency already discovered by PLANCK,
the case becomes very serious. It is clear that the properties of a
given ray when appearing as a rain of particles must be connected
with its properties when appearing as a train of waves. This is
indeed the case, and the connecting law is extremely simple when
all the particles of the beam have exactly the same velocity. Experi-
ment then shows that the corresponding train of waves has the
simplest form possible, which is called harmonic, and is charac-
terized by a definite sharp frequency and wave-length. The law
of PLANCK states that the kinetic energy of the panicles is exactly
proportional to the frequency of vibration of the wave; the factor
of proportionality, called PLANCK'S constant, and denoted by the
letter A, has a definite numerical value which is known from
experiment with fair accuracy.

There you have the logical difficulty: a particle with a given
velocity is, qua particle, a point, existing at any instant without
extension in space. A train of waves is by definition harmonic only
if it fills the whole of space and lasts from eternity to eternity!
[The latter point may not appear so evident; but a mathematical
analysis made by FOURIER more than a hundred years ago has
clearly shown that every train of waves finite in space and time
has to be considered as a superposition of many infinite harmonic
waves of different frequencies and wave-lengths which are arranged
in such a way that the outer parts destroy one another by inter-
ference; and it can be shown that every finite wave can be decom-
posed into its harmonic components.] BOHR has emphasized this
pouit by saying that PLANCK'S principle introduces an irrational
feature into the description of nature.

Indeed the difficulty cannot be solved unless we are prepared to
sacrifice one or other of those principles which were assumed as
fundamental for science. The principle to be abandoned now is
that of causality as it has been understood ever since it could be
formulated exactly. I can indicate this point only very shortly.
The laws of mechanics as developed by GALILEO and NEWTON allow
us to predict the future motion of a particle if we know its position
and velocity at a given instant. More generally, the future behaviour
of a system can be predicted from a knowledge of proper initial
conditions. The world from the standpoint of mechanics is an
automaton, without any freedom, determined from the beginning.
I never liked this extreme determinism, and I am glad that modern
physics has abandoned it. But other people do not share this view.

To understand how the quantum idea and causality are con-
nected, we must explain the second fundamental law relating

46 SOME PHILOSOPHICAL ASPECTS Otf MODERN PHYSICS

particles and waves. This can be readily understood with the help
of our example of the exploding mine and the machine-gun. If the
latter fires not only horizontally but equally in all directions, the
number of bullets, and therefore the probability of being hit, will
decrease with distance in exactly the same ratio as the surface of
the concentric spheres, over which the bullets are equally distri-
buted, increases. But this corresponds exactly to the decrease of
energy of the expanding wave of the exploding mine. If we now
consider light spreading out from a small source, we see immediately
that in the corpuscular aspect the number of photons will decrease
with the distance in exactly the same way as does the energy of
the wave in the undulatory aspect. I have generalized this idea for
electrons and any other kind of particles by the statement that we
have to do with 'waves of probability' guiding the particles in such
a way that the intensity of the wave at a point is always propor-
tional to the probability of finding a particle at that point. This
suggestion has been confirmed by a great number of direct and
indirect experiments. It has to be modified if the particles do not
move independently, but act on one another,- for our purpose,
however, the simple case is sufficient.

Now we can analyse the connection between the quantum laws
and causality.

Determining the position of a particle means restricting it
physically to a small part of space. The corresponding probability
wave must also be restricted to this small part of space, according to
our second quantum law. But we have seen that by FOURIER'S
analysis such a wave is a superposition of a great number of simple
harmonic waves with wave-lengths and frequencies spread over a
wide region. Using now the first quantum law stating the pro-
portionality of frequency and energy, we see that this geometrically
well-defined state must contain a wide range of energies. The
opposite holds just as well. We have derived qualitatively the
celebrated uncertainty law of HEISENBERG: exact determination
of position and velocity exclude one another; if one is determined
accurately the other becomes indefinite.

The quantitative law found by HEISENBERG states that for each
direction in space the product of the uncertainty interval of space
and that of momentum (equal to mass times velocity) is always the
same; being given by PLANCK'S quantum constant h.

Here we have the real meaning of this constant as an absolute
limit of simultaneous measurement of position and velocity. For
more complicated systems there are other pairs or groups of physical
quantities which are not measurable at the same instant.

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 47

Now we remember that the knowledge of position and velocity
at one given time was the supposition of classical mechanics for
determining the future motion. The quantum laws contradict this
supposition, and this means the break-down of causality and
determinism. We may say that these propositions are not just
wrong, but empty: the premise is never fulfilled.

The result that the discovery of the quantum laws puts an end
to the strict determinism which was unavoidable in the classical
period is of great philosophical importance by itself. After relativity
has changed the ideas of space and time, another of KANT'S cate-
gories, causality, has to be modified. The a priori character of these
categories cannot be maintained. But of course there is not a
vacuum now where these principles were previously; they are
replaced by new formulations. In the case of space and time these
are the laws of the four-dimensional geometry of MINKOWSKI. In
the case of causality there also exists a more general conception,
that of probability. Necessity is a special case of probability; it is a
probability of one hundred per cent. Physics is becoming a funda-
mentally statistical science. The mathematical theory called quan-
tum mechanics which expresses these ideas in a precise form is a
most wonderful structure, not only comparable with, but superior
to, classical mechanics. The existence of this mathematical theory
shows that the whole structure is logically coherent. But this proof
is rather indirect, and convincing only for those who understand
the mathematical formalism. It is therefore an urgent task to show
directly for a number of important cases why, in spite of the use of
two such different pictures as particles and waves, a contradiction
can never arise. This can be done by discussing special experimental
arrangements with the help of HEISENBERG'S uncertainty relation.
In complicated cases this sometimes leads to rather puzzling and
paradoxical results, which have been carefully worked out by
HEISENBERG, BOHR and DARWIN, my predecessor in this Chair.

I shall mention only one case. Looking -through a microscope I
can see a microbe and follow its motion. Why should it not be
possible to do the same with atoms or electrons, simply by using
more powerful microscopes? The answer is that looking through'
the microscope means sending a beam of light, of photons, through
it. These collide with the particles to be observed. If these are
heavy like a microbe or even an atom they will not be essentially
influenced by the photons, and the deflected photons collected by
the lenses give an image of the object. But if this is an electron,
which is very light, it will recoil on colliding with the photon, an
effect first directly observed by COMPTON. The change of velocity

48 SOME PHILOSOPHICAL ACf& Of

of the electron is to some extent indeterminate, and depends ori
the physical conditions in such a way that HEISENBERG'S uncertainty
relation is exactly fulfilled in this case also.

BOHR has introduced the expression 'complementarity' for the
two aspects of particles and waves. Just as all colours which we see
can be arranged in pairs of complementary colours giving white
when mixed, so all physical quantities can be arranged in two
groups, one belonging to the particle aspect, the other to the wave
aspect, which never lead to contradictions, but are both necessary
to represent the full aspect of nature.

Such a short expression for a complicated and difficult situation
is very useful, for instance, with respect to the naive question:
Now, what is a beam of light or a material substance 'really,' a
set of particles or a wave? Anybody who has understood the
meaning of complementarity will reject this question as too much
simplified and missing the point. But this rejection does not solve
the problem whether the new theory is consistent with the idea of
an objective world, existing independently of the observer. The
difficulty is not the two aspects, but the fact that no description of
any natural phenomenon in the atomistic domain is possible without
referring to the observer, not only to his velocity as in relativity,
but to all his activities in performing the observation, setting up
the instruments, and so on. The observation itself changes the order
of events. How then can we speak of an objective world ?

Some theoretical physicists, among them DIRAG, give a short and
simple answer to this question. They say: the existence of a mathe-
matically consistent theory is all we want. It represents everything
that can be said about the empirical world; we can predict with its
help unobserved phenomena, and that is all we wish. What you
mean by an objective world we don't know and don't care.

There is nothing to be objected against this stand-point except
one thing, that it is restricted to a small circle of experts. I cannot
share this I' art pour I 9 art standpoint. U think that scientific results
should be interpreted in terms intelligible to every thinking man.
To do this is precisely the task of natural philosophy.

The philosophers to-day concentrate their interest on other
questions, more important for human life than the troubles arising
from a refined study of atomistic processes. Only the positivists,
who claim to have a purely scientific philosophy, have answered
our question. Their standpoint (JORDAN, 1936) is even more radical
than that of DIRAG mentioned above. Whereas he declares himself
content with the formulae and uninterested in the question of an
objective world, positivism declares the question to be meaningless.

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 49

Positivism considers every question as meaningless which cannot
be decided by experimental test. As I said before, this standpoint
has proved itself productive by inducing physicists to adopt a critical
attitude towards traditional assumptions, and has helped in the
building of relativity and quantum theory. But I cannot agree
with the application made by the positivists to the general problem
of reality. If all the notions we use in a science had their origin in
this science, the positivists would be right. But then science would
not exist. Although it may be possible to exclude from the internal
activity of science all reference to other domains of thinking, this
certainly does not hold for its philosophical interpretation. The
problem of the objective world belongs to this chapter.

Positivism assumes that the only primary statements which are
immediately evident are those describing direct sensual impressions.
All other statements are indirect, theoretical constructions to
describe in short terms the connections and relations of the primary
experiences. Only these have the character of reality. The secondary
statements do not correspond to anything real, and have nothing
to do with an existing external world; they are conventions invented
artificially to arrange and simplify 'economically* the flood of
sensual impressions.

This standpoint has no foundation in science itself; nobody can
prove by scientific methods that it is correct. I would say that its
origin is metaphysical were I not afraid of hurting the feelings of the
positivists, who claim to have an entirely unmetaphysical philosophy.
But I may safely say that this standpoint rests on psychology, only
it is not a sound psychology. Let us consider it applied to examples
of everyday life. If I look at this table or this chair I receive innu-
merable sense-impressions patches of colour and when I move
my head these impressions change. I can touch the objects and
get a great variety of new sense-impressions, of varying resistance,
roughness, warmth, and so on. But if we are honest, it is not these
un co-ordinated impressions that we observe, but the total object
'table' or 'chair.' There is a process of unconscious combination,
and what we really observe is a totality which is not the sum of the
single impressions, not more or less than this sum, but something
new. What I mean will perhaps become clearer if I mention an
acoustical phenomenon. A melody is certainly something else than
the sum of the tones of which it is composed; it is a new entity.

Modern psychology is fully aware of this fact. I allude to the
Gestalt-psychology of v. EHRENFELS, KOHLER, and WERTHEIMER.
The word Gestalt, which seems to have no adequate English transla-
tion, means not only shape, but the totality which is really

50 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

perceived* I cannot explain it better than by referring again to the
example of melody. These Gestalten are formed unconsciously;
when they are considered by the conscious mind they become
conceptions and are provided with words. The unsophisticated
mind is convinced that they are not arbitrary products of the mind,
but impressions of an external world on the mind. I cannot see any
argument for abandoning this conviction in the scientific sphere.
Science is nothing else than common sense applied under unac-
customed conditions. The positivists say that this assumption of
an external world is a step into metaphysics, and meaningless,
since we can never know anything about it except by the percep-
tions of our senses. This is evident. KANT has expressed the same
point by distinguishing between the empirical thing and the 'thing
in itself' (Ding an sick] which lies behind it. If the positivists go on
to say that all our assertions regarding the external world are only
symbolical, that their meaning is conventional, then I protest.
For then every single sentence would be symbolical, conventional;
even if I merely say, 'Here I am sitting on a chair.' The 'chair' is
no primary sensual impression, but a notion connected with a
Gestalt, an unconscious integration of the impressions to a new
unit which is independent of changes in the impressions. For if I
move my body, my hands, my eyes, the sensual impressions change
in the most complicated way, but the e chair s remains. The chair is
invariant with respect to changes of myself, and of other things or
persons, perceived as Gestalten. This fact, a very obtrusive fact,
of 'invariance' is what we mean by Saying that there is 'really' a
chair. It can be submitted to test, not by physical experiment, but
by the wonderful methods of the unconscious mind, which is able to
distinguish between a 'real' and a painted chair by merely moving
the head a little. The question of reality is therefore not meaning-
less, and its use not merely symbolic or conventional.

The expression 'invariant' which I have already used in speaking
of relativity, and which appears here in a more general sense, is
the link connecting these psychological considerations with exact
science. It is a mathematical expression first used in analytical
geometry to handle quantitatively spatial Gestalten, which are
simple shapes of bodies or configurations of such. I can describe
any geometrical form by giving a sufficient number of co-ordinates
of its points; for instance, the perpendicular projections of its points
on three orthogonal co-ordinate planes. But this is by far too much;
it describes not only the form but the position relative to the three
arbitrary planes, which is entirely irrelevant. Therefore one has to
eliminate all the superfluous, uninteresting parts of the co-ordinate

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 51

description by well-known mathematical processes; the result is
the so-called invariants describing the intrinsic form considered.

Exactly the same holds if we have to do not only with size and
shape, but also with colour, heat, and other physical properties.
The methods of mathematical physics are just the same as those
of geometry, starting with generalized co-ordinates and eliminating
the accidental things. These are now not only situation in space,
but motion, state of temperature or electrification, and so on.
What remains are invariants describing things.

This method is the exact equivalent of the formation of Gestalten
by the unconscious mind of the man unspoiled by science. But
science transcends the simple man's domain by using refined
methods of research. Here unknown forms are found, for which the
unconscious process does not work. We simply do not know what
we see. We have to think about it, change conditions, speculate,
measure, calculate. The result is a mathematical theory repre-
senting the new facts. The invariants of this theory have the right
to be considered as representations of objects in the real world.
The only difference between them and the objects of everyday life
is that the latter are constructed by the unconscious mind, whereas
the objects of science are constructed by conscious thinking. Living
in a time in which FREUD'S ideas about the unconscious sphere are
generally accepted, there seems to be no difficulty in considering
this difference between common and scientific objects as of second
order. This is also justified by the fact that the boundary between
them is not at all sharp, and is continually changing. Conceptions
which once were purely scientific have become real things. The
stars were bright points on a spherical shell for the primitive man.
Science discovered their geometrical relations and orbits. It met
with furious opposition; GALILEO himself became a martyr to truth.
To-day these mathematical abstractions are common knowledge of
school-children, and have become part of the unconscious mind
of the European. Something similar has happened with the con-
ceptions of the electromagnetic field.

This idea that the invariant is the link between common sense
and science occurred to me as quite natural. I was pleased when
I found the same idea in the presentation of the Philosophy of Mathe-
matics by HERMANN WEYL (1926), the celebrated Princeton mathe-
matician. I think it is also in conformity with BOHR'S (1933) ideas.
He insists on the point that our difficulties in physics come from the
fact that we are compelled to use the words and conceptions of
everyday life even if we are dealing with refined observations. We
know no other way of describing a motion than either by particles

52 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

or waves. We have to apply them also in those cases where observa-
tion shows that they do not fit completely, or that we really have
to do with more general phenomena. We develop mathematically
the invariants describing the new observations, and we learn step
by step to handle them intuitively. This process is very slow, and
it proceeds only in proportion as the phenomena become known in
wider circles. Then the new conceptions sink down into the uncon-
scious mind, they find adequate names, and are absorbed into the
general knowledge of mankind.

In quantum theory we are only at the beginning of this process.
Therefore I cannot tell you in a few words of ordinary language
what the reality is which quantum mechanics deals with. I can
only develop the invariant features of this theory and try to describe
them in ordinary language, inventing new expressions whenever a
conception begins to appeal to intuition. This is what teaching of
physics means. Well-trained youth takes things for granted which
seemed to us horribly difficult, and later generations will be able
to talk about atoms and quanta as easily as we are able to talk
about this table and this chair, and about the stars in heaven. I do
not, however, wish to belittle the gap between modern and classical
physics. The idea that it is possible to think about the same pheno-
mena with the help of two entirely different and mutually exclusive
pictures without any danger of logical contradiction is certainly new
in science. BOHR has pointed out that it may help to solve funda-
mental difficulties in biology and psychology. A living creature,
plant, or animal is certainly a physico-chemical system. But it is
also something more than this. There are apparently two aspects
again. The time of materialism is over; we are convinced that the
physico-chemical aspect is not in the least sufficient to represent
the facts of life, to say nothing of the facts of mind. But there is the
most intimate connection between both spheres; they overlap and
are inter-woven in the most complicated way. The processes of life
and mind need other conceptions for their description than the
physico-chemical processes with which they are coupled. Why do
these differing languages never contradict each other? BOHR has
suggested the idea that this is another case of complementarity,
just as between particles and waves in physics. If you want to study
a specific biological or psychological process by the methods of
physics and chemistry, you have to apply all kinds of physical
apparatus, which disturbs the process. The more you learn about
the atoms and molecules during the process, the less you are sure
that the process is that which you want to study. By the time you
know everything about the atoms, the creature will be dead. This

SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS 53

is briefly BOHR'S suggestion of a new and deeper complementary
relation between physics and life, body and mind.

The old desire to describe the whole world in one unique philo-
sophical language cannot be fulfilled. Many have felt this, but to
modern physics belongs the merit of having shown the exact logical
relation of two apparently incompatible trends of thought, by
uniting them into a higher unit.

But with this result physics has not come to rest. It is the achieve-
ment of a bygone period, and new difficulties have appeared since.
Observations on nuclei, the innermost parts of the atoms, have
revealed a new world of smallest dimensions, where strange laws
hold. It has been shown that every kind of atom has a nucleus of
definite structure, consisting of a very close packing of two kinds of
particles, called protons and neutrons. The proton is the nucleus
of the lightest atom, hydrogen, with a positive electric charge.
The neutron is a particle of nearly the same weight, but uncharged.
In the atom the nucleus is surrounded by a cloud of electrons,
which we have mentioned several times. They are particles nearly
2,000 times lighter than the proton or the neutron; they carry a
negative charge equal and opposite to that of the proton. But
recently positive electrons or 'positrons' have also been discovered;
in fact, their existence was predicted by DIRAC on account of
theoretical considerations. Hence we have four kinds of particles,
two 'heavy' ones, proton and neutron, and two e light' ones, the
negative and positive electron, which can all move with any
velocity less than that of light. But then there are the photons,
which can move only with the velocity of light, and very likely
another kind of particles called 'neutrinos' the motion of which is
restricted in the same say.

The question which modern physics raises is: Why just these
particles ? Of course a question put like this is rather vague, but it
has a definite meaning. There is, for instance, the ratio of the masses
of proton and electron, the exact value of which has been found
to be 1845. Then there is another dimensionless number, 137, con-
necting the elementary charge, PLANCK'S quantum constant, and
the velocity of light. To derive these numbers from theory is an
urgent problem only a theory of this kind does not exist. It would
have to deal with the relations between the four ultimate particles.
There has been made the fundamental discovery that a positive
and a negative electron can unite to nothing, disappear, the energy
liberated in this process being emitted in the form of photons; and
vice versa, such a pair can be born out of light. Processes of this
type, transformations of ultimate particles including birth and

54 SOME PHILOSOPHICAL ASPECTS OF MODERN PHYSICS

death, seem to be the key to a deeper understanding of matter. We
can produce these violent processes in the laboratory only on a very
small scale, but nature provides us with plenty of material in the
form of the so-called cosmic rays. In observing them we are witnesses
of catastrophes in which by the impact of two particles large groups
of new particles are generated, which have received the suggestive
name of e showers.' We seem here to be at the limit where the con-
ception of matter as consisting of distinct particles loses its value,
and we have the impression that we shall have to abandon some
other accepted philosophical principle before we shall be able to
develop a satisfying theory.

It would be attractive to analyse the indications which our
present knowledge yields. But my time is over.

The purpose of my lecture has been to show you that physics,
besides its importance in practical life, as the fundamental science
of technical development, has something to say about abstract
questions of philosophy. There is much scepticism to-day about
technical progress. It has far outrun its proper use in life. The
social world has lost its equilibrium through the application of
scientific results. But Western man, unlike the contemplative
Oriental, loves a dangerous life, and science is one of his adventures.

We cannot stop it, but we can try to fill it with a true philosophical
spirit: the search of truth for its own sake.

REFERENCES

BOHR, N. (1933) 'Licht und Leben,' Naturwissenschqften, si, 245.
JORDAN, P. (1936) A brilliant presentation of the positivistic standpoint

is given in his book Anschauliche Quantentheorie, J. Springer, Berlin.
WEYL, H. (1926) 'Philosophic der Mathematik und Naturwissenschaft,'

Handbuch der Philosophic, Abt. II, A, n. Revised English version:

Philosophy of Mathematics and Natural Science, Princetown: University

Press, 1949.

CAUSE, PURPOSE AND ECONOMY IN

NATURAL LAWS
[MINIMUM PRINCIPLES IN PHYSICS]

[A lecture given at the Royal Institution of Great Britain Weekly Evening
Meeting, 10 February 1939. First published in Proc. Roy. Inst., Vol. XXX,
Part iii, 1939.]

WITHOUT claiming to be a classical scholar I think that the
earliest reference in literature to the problems which I wish to
treat to-night is contained in Virgil's Aeneid, Book I, line 368, in the
words 'taurino quantum possent circumdare tergo'.

The story, as told at greater length by the later Greek writer
ZOSIAS, is this: Dido, sister of King Pygmalion of the Phoenician city
of Tyre, a cruel tyrant who murdered her husband, was compelled
to fly with a few followers and landed at the site of the citadel of
Carthago. There she opened negotiations with the inhabitants for
some land and was offered for her money only as much as she could
surround with a bull's hide. But the astute woman cut the bull's
hide into narrow strips, joined them end to end, and with this long
string encompassed a considerable piece of land, the nucleus of her
kingdom. To do this she had evidently to solve a mathematical
question the celebrated problem of Dido: to find a closed curve of
given circumference having maximum area.

Well, we do not know how she solved it, by trial, by reasoning or
by intuition. In any case the correct answer is not difficult to guess,
it is the circle. But the mathematical proof of this fact has only been
attained by modern mathematical methods.

In saying that the first appearance of this kind of problem in
literature is that quoted above I am not, of course, suggesting that
problems of minima and maxima had never occurred before in the
life of mankind. In fact nearly every application of reason to a definite
practical purpose is more or less an attempt to solve such a problem;
to get the greatest effect from a given effort, or, putting it the other
way round, to get a desired effect with the smallest effort. We see
from this double formulation of the same problem that there is no
essential distinction between maximum and minimum; we can speak
shortly of an extremum and extremal problems. The business man uses
the word 'economy 9 for his endeavour to make the greatest profit out
of a given investment, or to make a given profit out of the least

55

56 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

investment. The military commander tries to gain a certain strate-
gical position with the minimum loss to his side, and maximum loss
to the enemy a procedure described by the experts by the dubious
expression 'economy of life'. These examples show how extremal
problems depend on ideas taken from human desires, passions,
greeds, hatreds; the ends to be achieved are often utterly unreason-
able, but once they are accepted as ends they lead to a strictly
rational question, to be answered by logical reasoning and mathe-
matics. Our whole life is just this mixture of sense and nonsense, to
attain by rational methods aims of doubtful character. Consider
our road system: does it meet the simple requirement of providing
the shortest connections between inhabited centres ? Certainly not.

Ax Ax

FIG. i . Maxima, Minima, Points of Inflexion.

The roads are the more or less rational resultant of geographical,
historical and economic conditions,which are often anything but
rational.

But here we have to do not with the activities of mankind but
with the laws of nature. The idea that such laws exist and that they
can be formulated in a rational way is a comparatively late fruit of
the human intellect. The nations of antiquity developed only a few
branches of science, notably geometry and astronomy, both for
practical purposes. Geometry arose from the surveying of sites and
from architecture, astronomy from the necessities of the calendar
and navigation.

Modern science began with the foundation of mechanics by
GALILEO and NEWTON. The distinctive quality of these great
thinkers was their ability to free themselves from the metaphysical
traditions of their time and to express the results of observations

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 57

and experiments in a new mathematical language regardless of any
philosophical preconceptions. Although NEWTON was a great theolo-
gian his dynamical laws are free from the idea that the individual
motion of a planet might bear witness to a definite and detectable
purpose. But during his lifetime, at the end of the seventeenth
century, geometrical and analytical problems of extremals began to
interest mathematicians, and shortly after NEWTON'S death hi 1727
the metaphysical idea of purpose or economy in nature was linked
up with them.

t"

2 V 6 7 8 10 12 1V 16 18 20 22 2*\
FIG. 2. Diurnal variation of temperature.

Before I go on to speak of the historical development, let us briefly
review those geometrical problems exemplified by Dido's land
purchase from which we started.

The top of a mountain, the bottom of a valley, are the prototypes
of maxima and minima; a vertical profile of a mountain range, as
shown in Fig. i, represents the simplest mathematical figure with
extremal points and we see that the tangent line is horizontal at
these points. As the figure shows, there are other points with horizontal
tangent, but the tangent is a so-called inflexional tangent. The com-
mon property of these points is that the height is stationary in their
neighbourhood; it does not change appreciably as it would if the
point were on a slope.

You will be acquainted with the method of graphs, representing
the law of change of any quantity by a curve on co-ordinate paper.
The diurnal variation of temperature, for instance, is shown by a

58 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

graph like this (Fig. 2); it shows a maximum shortly after noon,
and a minimum in the small hours of the morning.

Let us assume that Dido wished to build on her ground a rect-
angular building with an area as large as possible; this would mean

FIG. 3. Rectangles of equal circumference and
different area.

Sides
4x4
3 X5
2X6
i x 7

CiTCUfnference

2 x (4 + 4

2 x (3 +

2 X (2 +

2 X (I +7

16
= 16
= 16
= 16

Area
16
15

12

7

a modification, in fact a great simplification of her problem, as
she would not have to choose the curve of maximum area out of
all possible closed curves of given length, but merely the rectangle
of maximum area out of all rectangles of given circumference. Fig. 3
shows a set of such rectangles which have obviously all a smaller
area than the square.

FIG. 4. Rectangles of equal area and different circumference.

(A
(B
(C
(D

Sides Area Circumference

4x4 16 2 x (4 + 4) = 16

3 X 5*34 16 2 x (3 + 5-34) = 16-7

2x8 16 2 x (2 + 8) = 20

i X 16 16 2 x (i + 16) = 34

This is the simplest form of the genuine isoperimetric problem (from
the Greek: iso = equal, perimeter = circumference), the general
case of which is Dido's problem. But mathematicians nowadays use
this name for all kinds of problems in which an extremum has to
be determined under a constraining condition (as, for instance,
maximum area for given circumference). Here one can generally

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 59

interchange the two quantities concerned, whereby a maximum of
the one becomes a minimum of the other; the square, for instance,
is clearly also the rectangle of minimum length surrounding a
given area (Fig. 4) and the corresponding fact holds for the circle
as compared with all other closed curves.

FIG. 5.

Another type of problem is that connected with the idea of the
shortest line. The simplest case is that of choosing the point Q, on a
straight line L such that the distance from a given point P outside
the line may be as short as possible (Fig. 5). It is evident that Q,is
the foot of the normal from P to the line L. A little more involved
is the question how to find a point Q/on a straight line L so that
the sum of its distances PiQ, + QP a from two external points P x ,

Or

FIG. 6a.

P 2 is as small as possible. If P 1? P 2 are on different sides of the line L
the solution is trivial, namely, Q, is the point of intersection of L
\vith the straight line P^P 2 (Fig. 6a). But if P x and P ? are on the
same side of L the solution can easily be found by noticing that to
each point P 2 there belongs an 'image' point P 2 ' on the other side
of L, and Q,will be the intersection of P X P 2 ' with L (Fig. 6b). This
idea of an image presents the first example of a physical interpreta-
tion of such a geometrical problem. For it is evident that if L were
a plane mirror a beam of light travelling from P x to the mirror and

6o CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

reflected to P 2 would just coincide with our solution. This solution
is exactly the optical law of reflection, and we have expressed this as
a minimum principle: The beam of light selects just that reflecting
point Q, which makes the total path P X Q, + QP 2 as short as possible.
I have here a mechanical model to show this: the point Q,is repre-
sented by a little peg movable along a bar, and the beam of light
by a string fixed at one end at P 1? while the other end is in my hand.
If I pull the string you see that the point Q, adjusts itself so that
PiQ,, P 2 Q, make equal angles with the line, in agreement with the
image construction. The light behaves as if each beam had a
tendency to contract, and the French philosopher FERMAT has
shown that all the laws of geometrical optics can be reduced to the
same principle. Light moves like a tired messenger boy who has to
reach definite destinations and carefully chooses the shortest way

FIG. 7.

possible. Are we to consider this interpretation as accidental, or are
we to see in it a deeper metaphysical significance? Before we can
form a judgment we must learn more about the facts and consider
other cases.

Let us return to geometrical examples. So far we have assumed
that only straight line connections between different points are
admitted, or lines composed of straight parts (as in the last example).
But this restriction is not necessary, and if it is dropped we approach
the domain of problems to which the real Dido problem belongs,
namely those where a whole curve has to be determined from the
condition that some quantity shall be extremal.

The simplest question of this type is: why is the straight line the
shortest connection between two given points A and B? (Fig. 7). We
are here in a much higher branch of mathematics, in the realm of
infinite possibilities, called the calculus of variations. For we have to
compare the length of all possible curves passing through A and B,
that is an infinite number of objects which are not points, but
figures. It -is one of the great triumphs of the human mind to have

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 6 1

developed methods for performing this apparently superhuman
task.

If we travel on our earth we can never go exactly in a straight
line since the earth's surface is not plane. The best we can do is to
follow a great circle, which is the curve in which the sphere is inter-
sected by a plane passing through the centre. Indeed, it can be
shown that the shortest path between any two points A, B, not being
the ends of the same diameter ('antipodes'), is the arc of the great
circle through A and B, or better, the shorter of the two arcs. Ships
on the ocean should travel on great circles.

You know that the globe is not an exact sphere but is slightly
flattened at the poles, bulging at the equator. What, then, about the
shortest line on such a surface?

It is just about a hundred years ago that the great mathematician,
KARL FRIEDRICH GAUSS, in Gottingen, hit on this problem when
occupied with a geodetic triangulation of his country, the Electorate
of Hanover. As he was not merely a surveyor but one of the greatest
thinkers of all times, he attacked the problem from the most general
standpoint and investigated the shortest lines on arbitrary surfaces.
But in remembrance of his starting point, he called these lines
geodesies. I wish to say a few words about these lines and their
properties, as they are in many ways of fundamental importance
for physics.

GAUSS' investigation led him to the discovery of non-euclidean
geometry. This discovery is generally attributed to the Russian
LOBATSCHEFSKY and the Hungarian BOLYAI, and this is quite
correct, as these investigators published independently (about 1830)
the first systems of non-euclidean geometry. But the discovery (1899)
of GAUSS' diary many years after his death and the collection and
publication of his correspondence have given ample evidence that
a great number of the important mathematical discoveries made by
others during the first part of the eighteenth century were already
known to him, among them a complete theory of non-euclidean
space. He did not publish it because, as he wrote to a friend, he
was afraid e of the clamour of the Boeotians'. The proof that it is
possible to construct geometries differing from that of EUCLID
without meeting contradictions was a fundamental step towards the
modern development of science. It led to an empirical interpretation
of geometry as that part of physics which deals with the general
properties of the form and position of rigid bodies. Through the
work of RIEMANN and EINSTEIN, geometry and physics gradually
amalgamated to form a unity. But besides these important develop-
ments, the study of geodesies teaches us other things which throw

6'2 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

light on the character of different types of physical laws, and on
our subject of cause, purpose and economy in nature.

Let us consider a point P on a surface (Fig. 8) and all curves
through P which have the same direction at P. It is evident that
there is among them a e straightest curve', i.e. one with the smallest
curvature. I have a model of a surface with the help of which I can
.demonstrate to you the straightest curve. There are two small loops
fixed on the surface, through which I can thread a piece of a piano
wire. This offers resistance to bending in virtue of its elastic proper-
ties and, therefore, assumes the straightest shape possible on the
surface. I now take a piece of string and pull it through the two
loops. This, of course, assumes the shape of the shortest connection
between the two points possible on the surface. You see that the
straightest line and the shortest line coincide accurately.

FIG. 8. Lines of minimum curvature on a surface.

Hence the geodesic can be characterized by two somewhat
different minimum properties : one which can be called a local or
differential property, namely, to be as little curved as possible at a
given point for a given direction; and the other, which can be called
total or integral, namely, to be the shortest path between two points
on the surface.

This dualism between 'local' and 'total' laws appears not only
here in this simple geometrical problem, but has a much wider
application in physics. It lies at the root of the old controversy
whether forces act directly at a distance (as assumed in NEWTON'S
theory of gravitation and the older forms of the electric and magnetic
theories), or whether they act only from point to point (as in
FARADAY'S and MAXWELL'S theory of electromagnetism and all
modern field theories). We can illustrate this by interpreting the
law of the geodesic itself as a law of physics, in particular of dyna-
mics. NEWTON'S first law of dynamics, the principle of inertia, states
that the straight line is the orbit of any small particle moving free

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 63

from external forces; a billiard ball moves in a straight line if the
table is accurately horizontal so that gravity is inoperative. Imagine
a frozen lake so large that the curvature of the earth is perceptible
over its length there is no straight line on it, only straightest lines,
the great circles of the globe. It is clear that these are the orbits of
free particles. We can, therefore, extend NEWTON'S first law to the
motion on smooth surfaces by saying that a body free from external
forces travels as straight as possible. Here we have a physical law
of the local character. But, knowing the other minimum property of
the geodesic, we can also say: a body always moves from one
position to any other by the shortest possible path which is a law
of the integral type.

FIG. 9 (a). Catenary, (b). Chain of four elements carrying a con-
struction \vhich makes the centre of gravity visible.

There seems to be no objection to extremal laws of the local type,
but those of the integral type make our modern mind feel uneasy.
Although we understand that the particle may choose at a given
instant to proceed on the straightest path we cannot see how it can
compare quickly all possible motions to a distant position and choose
the shortest one this sounds altogether too metaphysical.

But before we follow out this line of thought we must convince
ourselves that minimum properties appear in all parts of physics,
and that they are not only correct but very useful and suggestive
formulations of physical laws.

One field in which a minimum principle is of unquestionable
utility is statics, the doctrine of the equilibrium of all kinds of systems
under any forces. A body moving under gravity on a smooth
surface is at rest in stable equilibrium at the lowest point, as this
pendulum shows. If we have a system composed of different bodies
forming a mechanism of any kind, the centre of gravity tends to
descend as far as possible; to find the configuration of stable equi-
librium one has only to look for the minimum of the height of the

64 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

centre of gravity. This height, multiplied by the force of gravity, is
called potential energy.

A chain (Fig. 9 (a) ) hanging from both ends assumes a definite shape,
which is determined by the condition that the height of the centre
of gravity is a mininium. If the chain has very many links, we get
a. curve called the catenary. We have here a genuine variational
problem of the isoperimetric type, for the catenary has the lowest
centre among the infinite variety of curves of the same length
between the given end-points. I have here a chain consisting of only
four links (Fig. 9(b)). The centre of gravity is made visible by a

FIG 10. Steel tape carrying a weight (Elastica).

construction of levers (made from light material so that they do not
contribute appreciably to the weight). If I disturb the equilibrium
of the chain in an arbitrary way you observe that the centre of
gravity is always rising.

I will now show you an example where gravity competes with
another force, elasticity (Fig. i o) . I have chosen this special problem,
not because it was the subject of my doctor's thesis more than 30 years
ago, but because it can be used to explain the difference between
the genuine minimum principles of statics and the formal variational
principles of dynamics, as we shall see later on. A steel tape is
clamped at one end and carries a weight at the other. This weight
is pulled downwards by gravity, while the tape tries to resist bending
in virtue of its elasticity. This elastic force also has a potential

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 65

energy; for a definite amount of work must be done to bend the
tape into a given curved shape, and it is clear that this energy
depends in some way on the curvature of the tape which varies
from point to point. Now there is a definite position of equilibrium,
which you see here, namely a position in which the total energy,
that of gravitation plus that of elasticity, is as small as possible; if I
pull the weight down the gravitational energy decreases, but the
energy of elastic bending increases more, so that there results a
restoring force; and if I lift the weight, the gravitational energy
increases more than the bending energy decreases so that the force
is again in the direction towards equilibrium. You see that for some
directions of the clamped end there are two positions of equilibrium,
one on the left and one on the right.

This also holds for vertical clamping where the two equilibrium
forms are symmetrical but only if the tape is long enough. If I
shorten its length sufficiently, the only possible equilibrium form
is that in which the tape is straight. There is a definite length for a
given weight at which this straight form becomes unstable: it is
determined by the condition that beyond this length the potential
energy ceases to be a minimum for the straight form and becomes a
minimum for a curved form.

The formula for this characteristic length was found by EULER
and plays an important role in engineering, as it determines the
strength of vertical bars and columns. But similar instabilities also
occur for inclined directions of clamping. If I fix the length and
change the clamping angle, a jump suddenly occurs from one
position to the one on the opposite side. This instability is again
determined by the condition of minimum energy. We can summarize
the facts connected with the limits of stability by drawing a graph,
not of the elastic lines themselves (which are beautiful curves like
those shown in Fig. 1 1, called elastica), but by plotting the angle of
inclination against the distance from the free end. We now obtain
wave-shaped curves (Fig. 12), all starting horizontally from the line
representing the end carrying the weight. You see that these curves
have an envelope and the calculation shows that this envelope is
just the limit of stability. Through any point on the right of the
envelope there pass at least two curves; this corresponds to the fact
that this point represents a clamping angle for which two equilibria
exist. If we now move vertically upwards in the diagram, we change
the angle of clamping (without changing the length of the tape) ;
when we cross the envelope we pass into a region where there is
only one curve through each point. At the envelope one of the con-
figurations becomes unstable and jumps across to the other one. In

66

CAUSE. PURPOSE AND ECONOMY IN NATURAL LAWS

particular EULER'S limit for the stability of the straight form of the
tape is represented by the sharp point of the envelope; the distance
of this from the origin is just a quarter of the wave length of the
neighbour curve, which value gives exactly EULER'S formula. I am
going to ask you to keep this example in mind as we shall return to
it later, when we discuss the minimum principles of dynamics.

t

N 1
x >

FIG. n. Elastica.

Another example of the statical principle of minimum energy is
provided by soap bubbles. Soap films have the property of contracting
as much as possible; the potential energy is proportional to the
surface-area. A well-known experiment shows this very clearly. I
project a soap film stretched over a wire in the form of a circle on
which a fine thread is fixed. If I destroy the film on one side of the
thread, the film on the other side contracts, the thread is pulled

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

6 7

tight and assumes- the form of an arc of a circle. I now take a closed
loop of thread; if I destroy the film in the inner portion the loop
immediately forms a perfect circle under the stress of the outer film,
showing that this film is under a uniform tension.

360*

FIG. 12. Diagram showing the limits of stability of the elastica.

It is clear, therefore, that a closed soap bubble filled with air and
floating freely in space has the shape of a sphere, which is the
minimum surface for a given volume the spatial analogue of
Dido's problem.

There exist other rninirnum surfaces not closed but determined
by a given boundary. We have only to bend a wire to the shape of

68 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

this boundary and to dip it into a soap solution to get a perfect
physical model of the minimum surface. These experiments and
their theory were studied long ago by the blind French physicist,
PLATEAU, and you will find a wonderful account of them in the
celebrated little book by G. V. BOYS on Soap Bubbles. I will show you
some of them. See how expert a mathematician Nature is and how
quickly she finds the solution !

Some of you may consider these experiments merely as pretty
toys without any serious background. But they are chosen only for
the sake of illustration. The real importance of the principle of
minimum energy can scarcely be exaggerated. All engineering
constructions are based on it, and also all structural problems in
physics and chemistry.

As an example, I shall show you here some models of crystal
lattices. A crystal is a regular arrangement of atoms of definite
kinds in space. The discovery of LAUE, FRJEDRICH and KNIPPING
that X-rays are diffracted by these atomic lattices was used by
Sir WILLIAM BRAGG and his son, Professor W. L. BRAGG, for the
empirical determination of the atomic arrangements. A great
number of these are now well known; for instance, here are two
simple models, each consisting of two kinds of atoms in equal
numbers per unit of the lattice, but different in structure. One is the
lattice of a salt, sodium chloride (NaCl), the other of a similar salt,
caesium chloride (CsCl). The question -arises, why are they dif-
ferent? The answer can be expected only from a knowledge of the
forces between the atoms; for it is clear that the structure is deter-
mined by the condition of minimum potential energy. Conversely,
a study of this equilibrium condition must teach us something about
the character of the atomic forces. I have devoted considerable
energy to research in this field; it could be shown that the forces in
all these salt crystals are mainly the electrostatic interactions be-
tween the atoms which are charged, but that the difference of
stability between the two lattice types has its origin in another
force, namely the universal cohesion which causes gases to condense
at low temperatures. This force, called VAN DER WAALS' attraction,
is larger for bigger atoms; and as the caesium atoms are much
larger than the sodium atoms the minimum of potential energy is
attained for different configurations in caesium and sodium salt.

Considerations of this kind, more or less quantitative, enable us
to understand a great number of facts about the internal structure of
solid matter.

Similar methods can also be applied to the equilibrium of atoms
in molecules, but I shall not discuss them, for the problem of atomic

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 69

structure is really not one of statics but dynamics, as it involves the
motion of electrons in the atom.

Before we proceed to the consideration of minimum principles in
dynamics where the situation is not as clear and satisfactory as in
statics, we must first mention another part of physics which in a
sense occupies an intermediate position between statics and dyna-.
mics. It is the theory of heat, thermodynamics and statistical mechanics.
The phenomena considered are of this type. Substances of different
composition and temperature are brought into contact or mixed
and the resultant system observed. We have, therefore, to do with
the transition from one state of equilibrium to another, but we are
not so much interested in the process itself as in the final result. I
have here a glass of water and a bottle containing a dye; now I
pour the red dye into the water and observe the resultant solution.
If we look for a mechanical process with which to compare these
processes the nearest is, I think, the elastic steel tape carrying a weight
which we have already considered. If one end is fixed vertically
there are two stable equilibria; the system can be made to jump
over from one to the other by imparting energy to it, but you see
that it jumps back again. The process is reversible, it leads to a
definite final equilibrium only if the superfluous energy is taken
away. But in such a, case as that of the mixture of two liquids a final
equilibrium is automatically reached and the process is irreversible.
Not only does it never return spontaneously to the unmixed condi-
tion, but even the artificial separation of the dye from the water
cannot be performed by any simple means.

There is a very important extremum principle, discovered by
Lord KELVIN, which governs irreversible processes: A certain
quantity called entropy increases in the process and has a maximum
for the final equilibrium state. It is not easy to describe this
miraculous entropy in terms of directly observable quantities, such
as volume, pressure, temperature, concentration, heat. But its
meaning is immediately obvious from the standpoint of atomic
theory. What happens if the red solution spreads in the pure water?
The molecules of the red dye, at first concentrated in a restricted
volume, spread out over a greater volume. A state with a higher
degree of order is replaced by one of less order. To explain this
expression I have here a model, a flat box, like a little billiard table,
into which I can put marbles (purchased at Woolworth's for six-
pence). If I place them carefully in the right-hand half, I have a
state of partial order; if I shake the box they spread out over the
whole box and attain a configuration of lower order. If I throw
20 marbles into the box one after the other so that their position is

70 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

purely accidental it is very improbable that they will all fall in the
right-hand half. One can easily calculate the probability of a uniform
distribution over the whole box as compared with one in which the
majority of the marbles is in the right half; and one finds over-
whelming odds in favour of the uniform distribution. Now the
statistical theory of heat interprets the entropy of a system with the
aid of the probability of the distribution of the atoms, and this helps
us to understand why entropy always increases and tends to a

maximum.

/AAAAA\

/A A A A A AY

YAAAAAAAY

/AAAAAAAAY

/AAAAAAAAAY

/AAAAAAAAAAV

FIG. 13. Gallon's quincunx.
(By courtesy of the Institution of Electrical Engineers)

To show you the working of probability I have here a machine
(Fig. 13), invented by GALTON and called the quincunx. Shot falls
from a hole in the centre of the upper end and strikes numerous
obstacles in the shape of narrow triangles. At each encounter the
probabilities of falling to the right and to the left are equal. It is
clear that a ball has very little chance of always being deflected in
the same direction; therefore the cells collecting the balls at the
bottom will be comparatively empty at the end, and fuller in the
middle. The middle cell corresponds to those balls which have been
deflected an equal number of times to the right and to the left, that
is to the uniform distribution of deflections. You see that there is
a clear maximum. This demonstrates the uniform distribution of
the marbles or of the red dye molecules.

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 71

The thermodynamical principle of maximum entropy is, there-
fore, really a statistical law and has very little to do with dynamics
at all. If a system is initially in a state of partial order, that is in a
state which is not the most probable one (which would correspond
to the middle cell of the quincunx) it is very probable that after a
while it will have approached the state of maximum probability
or maximum entropy. Very probable, indeed but not absolutely
certain. And the modern technique of micro-observations has
revealed cases where deviations from the most probable state are
detectable. The extremal principle of statistical mechanics is, there-
fore, somewhat different in character from the similar laws of pure
mechanics. But I cannot go more deeply into the difficult questions
of the role of chance and probability in science.

Straight line

FIG. 14. Brachistochrone.

Let us now come back to the minimum principles of dynamics.

The first problem of this kind first both in historical order and
hi order of simplicity was formulated at the end of the seventeenth
century by JOHANN BERNOULLI of Basle, one of a great family which
produced many famous scholars and especially many mathemati-
cians. It is the problem of the curve of quickest descent or brachisto-
chrone (Greek: brachys = short, chronos = time) : given two points
at different levels, not in the same vertical, to determine a connecting
curve in such a way that the time taken by a body to slide without
friction under the action of gravity from the higher point to the
lower is a minimum compared, of course, with all possible curves
through the two points. I have here a model illustrating this prob-
lem, but instead of an infinite number of curves I have only three,
a straight line, an arc of a circle, and an intermediate curve (Fig.
14). Instead of the bodies sliding without friction I use steel ball-
bearings rolling on two rails. This has the advantage not only of
diminishing friction, but also of retarding the whole motion, which

72 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

would be too fast without this precaution. As the distance between
the rails is a fraction of the diameter of the sphere, it advances for
each full rotation only a fraction of the distance it would advance
if rolling on a smooth surface. The effect of this trick is only to
increase the inertia without changing the gravitational force; the
laws of motion are unchanged, only the time scale is reduced.

Now, before I start a race between three balls I ask you to bet
if you like which ball will win, and I am prepared to act as book-
maker. It is, of course, not any actual virtue of the ball to be the
fastest but of the shape of the curve on which it is rolling.

You see that it is not the straight line which carries the winner,
nor the steep descent of the circular arc, but just the intermediate
curve. If you were to try with any other curves you would always
find the same result; for this curve has been constructed according
to the theoretical calculation. It is a so-called cycloid, a curve which
you can observe hundreds of times every day on the road. It is the
curve traced out by a point on the circumference of a wheel rolling
along a straight line; I have here a circular disc with a piece of
chalk attached to it and if I roll it along the blackboard you see the
chalk drawing this line.

The determination of this brachistochronic property of the cycloid
was a very satisfactory piece of mathematics; it is a genuine mini-
mum problem and its solution was a great achievement. It attracted
much attention and there is no philosopher of this period who did
not test his analytical powers by solving similar extremal problems.
Another member of the Bernoulli family, DANIEL BERNOULLI,
developed during the beginning of the eighteenth century the
minimum principle of statics which we have already treated, and
applied it to the catenary and the elastic line. Encouraged by these
successes DANIEL BERNOULLI raised the question whether it was
possible to characterize the orbit, and even the motion in the orbit,
of a body subject to given forces for example, a planet by a
minimum property of the real motion as compared with all other
imagined or virtual motions. He put this question to the foremost
mathematician of his time, LEONARD EULER, who was very much
interested in. it and spent several years in investigating it. In the
autumn of 1743 he found a solution which he explained with the
help of various examples in an appendix to a book on isoperimetric
problems published in 1744. It is the basis of the principle of least
action which has played so prominent a part in physics right up to
the present time. But the history of this principle is an amazing
tangle of controversies, quarrels over priority and other unpleasant
things. MAUPERTUIS, in the same year, 1744, presented a paper to

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 7$

the Paris Academy in which he substituted for FERMAT'S optical
principle of the shortest light path, which we have already discussed,
a rather arbitrary hypothesis and extended the latter, in 1746, to all
kinds of motions. He defined action^ following LEIBNIZ, as the product
of mass into the velocity and the distance travelled, and he put
forward the universal principle that this quantity is a minimum for
the actual motion. He never gave a satisfactory proof of his principle
(which is not surprising as it is incorrect) but defended it by meta-
physical arguments based on the economy of nature. He was
violently attacked, by CHEVALIER D'ARCY in Paris, SAMUEL KONIG
from Bern and others who showed that if MAUPERTUIS' principle
were true, thrifty nature would be forced in certain circumstances
to spend not a minimum but a maximum of action. EULER, whose
principle is quite correct, behaved rather strangely; he did not
claim his own rights but even expressed his admiration for MAU-
PERTUIS' principle which he declared to be more general. The
reasons for this attitude are difficult to trace. One of them seems to
be the publication by KONIG of a fragment of an alleged letter of
LEIBNIZ in which the principle was enunciated. The genuineness of
this letter could never be proved and it seems probable that it was
a forgery designed to weaken MAUPERTUIS' position. This may have
brought EULER over to the side of MAUPERTUIS who was at this
time President of the Berlin Academy and a special favourite of the
KING FREDERIC II, later known as the Great. The dispute was now
carried over into the sphere of the court of Sanssouci and even into
the arena of politics. VOLTAIRE, friend of Frederic, who heartily
disliked the haughty President of the Academy, took the side of the
'underdog', KONIG, and wrote a caustic pamphlet, c Dr. Akakia',
against MAUPERTUIS. But the King, although he thoroughly enjoyed
VOLTAIRE'S witty satire, could not sacrifice his grand President and
was compelled to defend MAUPERTUIS. This led at last to the
disruption of their friendship and to VOLTAIRE'S flight from Berlin,
as described in many biographies of Frederic and of VOLTAIRE.

The curse of confusion has rested for a long period on the principle
of least action. LAGRANGE, whose work was the culmination of the
development of NEWTON'S dynamics, gives an unsatisfactory formu-
lation of the principle. JAGOBI restricts it in such a way that the
minimum condition determines the orbit correctly; the motion in
the orbit must be found with the help of the energy equation. This
was an important step. But the spell was at last broken by the great
Irishman, Sir WILLIAM ROWAN HAMILTON, whose principle is
mathematically absolutely correct, simple and general. At the same
time it put an end to the interpretation of the principle expressing

74 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

the economy of nature. Let us look quite briefly at the real situation.

We have already considered the quantity called potential energy of
a set offerees, which is the amount of work which must be done to
bring the mechanical system into a given configuration and there-
fore represents a measure of the ability of the system to do work.
This potential energy depends only on the configuration and has its
minimum in the equilibrium position. If the system is in motion
part of the potential energy is converted into energy of motion or
kinetic energy, namely the sum of half the mass into the square of the
velocity of the particles. The law of conservation of energy states
that the sum of the two forms of energy is always constant. Now the
principle of Hamilton has to do not with the sum but with the
difference of these two kinds of energy. It states that the law of
motion is such that a quantity frequently called action, namely the
sum of the contributions of each time interval to the difference of
kinetic and potential energy, is stationary for the actual motion, as
compared with all virtual motions starting at a given time from a
given configuration and arriving at a given subsequent time at
another given configuration.

Purposely I say stationary, not minimum, for indeed there is in
general no minimum.

What really happens can be explained very clearly with the help
of the simple pendulum. For there is, by a kind of fortunate mathe-
matical coincidence, a statical problem for which the genuine
minimum principle for the potential energy coincides formally
with the principle of least action for the pendulum. This is our old
friend the steel tape. In fact, the sum of the bending energy of the
weight attached is exactly the same mathematical expression as the
total action of the pendulum (the sum of the contributions of all
time elements to the difference of kinetic and potential energy) ;
therefore, the curves representing the angle of inclination of the
elastic line as a function of the distance from the free end are exactly
the same lines as those representing the angle of deflection of the
pendulum as a function of time. You see the vibrational character
in the graph although only a small part of the curve is drawn.

Now we have seen that only those regions of the graph, which
are simply covered by the lines, correspond to a real minimum, a
stable configuration of the elastic line. There are other regions,
those beyond the envelope, where two or more lines pass a given
point. Only one of those lines corresponds to a real minimum. But
both represent possible motions of the pendulum. Although the
conditions at the ends of the elastic tape do not correspond exactly
to those at the ends of the time interval in HAMILTON'S principle,

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 75

there is this fact in common. If the length of the tape, or the corres-
ponding time interval in HAMILTON'S principle for the pendulum
exceeds a certain limit, there is more than one possible solution, and
not each of them can correspond to a true minimum, though to a
possible motion. In this way we come to the conclusion that the
actual motion is not in every case distinguished by a genuine
extremal property of action, but by the fact that the action is
stationary as explained at the beginning of the lecture.

Thus the interpretation in terms of economy breaks down. Jf
nature has a purpose expressed by the principle of least action it is
certainly not anything comparable with that of a business man. We
may, I think, regard the idea of finding purpose and economy in
natural laws as an absurd piece of anthropomorphism, a relic of a
time when metaphysical thinking dominated science. Even if we
accept the idea that nature is so thrifty with her stock of action that
she tried to save it as long as possible she succeeds, as we have
seen, only during the first small part of the motion we cannot help
wondering why she considers just this strange quantity as especially
valuable.

The importance of HAMILTON'S principle lies in a different direction
altogether.

It is not nature that is economical but science. All our knowledge starts
with collecting facts, but proceeds by summarizing numerous facts
by simple laws, and these again by more general laws. This process
is very obvious in physics. We may recall, for instance, MAXWELL'S
electromagnetic theory of light by which optics became a branch of
general electrodynamics. The minimum principles are a very power-
fill means to this end of unification. This is easily understood by
considering the simplest example, that of the shortest path. If a
military commander has a good map he can move his troops from
one given point to another by simply announcing the point of
destination, without caring much about the details of the route,
since he supposes that the officer of the detachment will always
march by the shortest route. This minimum principle, together
with the map, regulates all possible movements. In the same way
the minimum principles of physics replace innumerable special laws
and rules always supposing the map, or in this case the kinetic
and potential energy, are given.

The ideal would be to condense all laws into a single law, a
universal formula, the existence of which was postulated more than a
century ago by the great French astronomer LAPLACE.

If we follow the Viennese philosopher, ERNST MAGH, we must
consider economy of thought as the only justification of science. I

76 CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS

do not share this view; I believe that there are many other aspects
and justifications of science but I do not deny that economy of
thought and condensation of the results are very important, and I
consider LAPLACE'S universal formula as a legitimate ideal. There is
no question that the Hamiltonian principle is the adequate formula-
tion of this tendency. It would be the universal formula if only the
correct expressions for the potential energy of all forces were known.
Nineteenth century thinkers believed, more or less explicitly, in this
programme and it was successful in an amazing degree.

By choosing a proper expression for the potential energy nearly
all phenomena could be described, including not only the dynamics
of rigid and elastic bodies but also that of fluids and gases, as well

FIG. 15.

as electricity and magnetism, together with electronic theory and
optics. The culmination of this development was EINSTEIN'S theory
of relativity, by which the abstract principle of least action regained
a simple geometrical interpretation, at least that part of it depending
on the kinetic energy. For this purpose one has to consider time as a
fourth co-ordinate, as Fig. 15 shows (where one dimension of space
is omitted) ; a motion is then represented by a line in this 4-dimen-
sional world in which a non-euclidean geometry is valid, of the
type invented by RIEMANN. The length of this line between two
points is just the kinetic part of the action in HAMILTON'S principle,
and the lines representing motions (under the action of gravity) are
geodesies of the 4-dimensional space. EINSTEIN'S law of gravitation,
which contains NEWTON'S law as a limiting case, can also be derived
from an extremum principle in which the quantity which is an
extremum can be interpreted as the total curvature of the space-
time world. But these are abstract considerations on which I cannot
dwell here.

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 77

We call this period of physics, which ends with the theory of
relativity, the classical period in contrast to the recent period which
is dominated by the quantum theory.

The study of atoms, their decomposition into nuclei and electrons,
and the disintegration of the nuclei themselves has led to the convic-
tion that the laws of classical physics do not hold down to these
minute dimensions. A new mechanics has been developed which
explains the observed facts very satisfactorily but deviates widely
from classical conceptions and methods. It gives up strict deter-
minism and replaces it by a statistical standpoint. Consider as an
example the spontaneous disintegration of a radium atom; we
cannot predict when it will explode but we can establish exact laws
for the probability of the explosion, and therefore predict the
average effects of a great number of radium atoms. The new
mechanics assumes that all laws of physics are of this statistical
character. The fundamental quantity is a wave function which obeys
laws similar to those of acoustical or optical waves; it is not, how-
ever, an observable quantity but determines indirectly the prob-
ability of observable processes. The point which interests us here is
the fact that even this abstract wave function of quantum mechanics
satisfies an extremum principle of the Hamiltonian type.

We are still far from knowing LAPLACE'S universal formula but
we may be convinced that it will have the form of an extremal
principle, not because nature has a will or purpose or economy, but
because the mechanism of our thinking has no other way of
condensing a complicated structure of laws into a short expression.

APPENDIX

As the argument against the economic interpretation of the principle
of least action rests on the comparison of the dynamical problem of
the pendulum and the statical problem of the loaded elastic tape,
readers who know some mathematics may welcome a few formulae
showing the identity of the variational principles for these two
examples.

If / is the length of the string, and 6 is the angle of deflection,
(Fig. 1 5 A) then-j- is the angular and -7- the linear velocity; there-

(///5\
I 2 where m is the mass of the

bob. The height of the bob above its lowest position is, as the figure
shows, / I cos 6. Multiplying this by the weight mg (g acceleration

7& CAtSE, PUfcPOSE AND ECONOMY IN NATURAL LAWS

of gravity) we get the potential energy; but as a constant does not
matter we can omit mgl and write the potential energy U = mgl
cos 6. The difference of kinetic and potential energy is T U

= 772/2 j I* + mgl cos 6, and the action during the time interval
\dt)

i* ( fsifi\ ~\

from t = o to t = r is I -I \A ( ] + W cos 6 \ dt> where the
J I \dtj J

o

abbreviations A = ml 2 and W = mgl are used.

We now consider the elastic tape. The energy stored up in the
element ds of the tape by bending it into a curve of radius of

FIG 16.

curvature p is \A ds, where A is the bending modulus. The figure

(153) shows that ds = pd0 } so that the bending energy of ds is \A

i

( | 2 ds, and the total elastic energy of bending %A [ } ds 9
\dsj J \ds J

o

where I is the length of the tape.

The potential energy of the weight W attached to the end is Wh

where A is the height of this weight above the level of the clamped

end. The figure (150) shows that h consists of contributions cos

ds of the single elements of the tape; therefore the potential energy

/

of the weight is W cos 6 ds . Adding these two potential energies

o

we get

CAUSE, PURPOSE AND ECONOMY IN NATURAL LAWS 79

an expression which is identical with the action of the pendulum if
the element ds and the total length I of the tape are replaced by the
time element dt and the total time r in the case of the pendulum.

EINSTEIN'S STATISTICAL THEORIES

[First published in Vol. VII of The Library of Living Philosophers', Albert
Einstein: Philosopher-Scientist, 1949.]

ONE of the most remarkable volumes in the whole of scientific
literature seems to me Vol. 17 (4th series) ofAnnalen der Physik,
1905. It contains three papers by EINSTEIN, each dealing with a
different subject, and each to-day acknowledged to be a master-
piece, the source of a new branch of physics. These three subjects,
in order of pages, are: theory of photons, Brownian motion, and
relativity.

Relativity is the last one, and this shows that EINSTEIN'S mind at
that time was not completely absorbed by his ideas on space and
time, simultaneity and electro-dynamics. In my opinion he would
be one of the greatest theoretical physicists of all times even if he
had not written a single line on relativity an assumption for which
I have to apologize, as it is rather absurd. For EINSTEIN'S conception
of the physical world cannot be divided into watertight compart-
ments, and it is impossible to imagine that he should have by-passed
one of the fundamental problems of the time.

Here I propose to discuss EINSTEIN'S contributions to statistical
methods in physics. His publications on this subject can be divided
into two groups: an early set of papers deals with classical statistical
mechanics, whereas the rest is connected with quantum theory.
Both groups are intimately connected with EINSTEIN'S philosophy of
science. He has seen more clearly than anyone before him the
statistical background of the laws of physics, and he was a pioneer
in the struggle for conquering the wilderness of quantum phenomena.
Yet later, when out of his own work a synthesis of statistical and
quantum principles emerged which seemed to be acceptable to
almost all physicists, he kept himself aloof and sceptical. Many of
us regard this as a tragedy for him, as he gropes his way in lone-
liness, and for us who miss our leader and standard-bearer. I shall
not try to suggest a resolution of this discord. We have to accept the
fact that even in physics fundamental convictions are prior to
reasoning, as in all other human activities. It is my task to give an
account of EINSTEIN'S work and to discuss it from my own philoso-
phical standpoint.

EINSTEIN'S first paper of 1902, 'Kinetische Theorie des Warme-
gleichgewichtes und des zweiten Hauptsatzes der Thermodynamik' 1

80

EINSTEIN'S STATISTICAL THEORIES 81

is a remarkable example of the fact that when the time is ripe
important ideas are developed almost simultaneously by different
men at distant places. EINSTEIN says in his introduction that nobody
has yet succeeded in deriving the conditions of thermal equilibrium
and of the second law of thermodynamics from probability con-
siderations, although MAXWELL and BOLTZMANN came near to it.
WILLAKD GIBBS is not mentioned. In fact, EINSTEIN'S paper is a
re-discovery of all essential features of statistical mechanics and
obviously written in total ignorance of the fact that the whole
matter had been thoroughly treated by GIBBS a year before (1901).
The similarity is quite amazing. Like GIBBS, EINSTEIN investigates
the statistical behaviour of a virtual assembly of equal mechanical
systems of a very general type. A state of the single system is des-
cribed by a set of generalized co-ordinates and velocities, which
can be represented as a point in a 2n-dimensional 'phase-space'; the
energy is given as function of these variables. The only consequence
of the dynamical laws used is the theorem of LIOUVTLLE according
to which any domain in the 2w-dimensional phase-space of all co-
ordinates and momenta preserves its volume in time. This law
makes it possible to define regions of equal weight and to apply the
laws of probability. In fact, EINSTEIN'S method is essentially identical
with GIBBS' theory of canonical assemblies. In a second paper, of the
following year, entitled 'Eine Theorie der Grundlagen der Thermo-
dynamik,' 2 EINSTEIN builds the theory on another basis not used
by GIBBS, namely on the consideration of a single system in course
of time (later called ' Zeit-Gesamtheit? time assembly), and proves
that this is equivalent to a certain virtual assembly of many systems,
GIBBS' micro-canonical assembly. Finally, he shows that the cano-
nical and micro-canonical distribution lead to the same physical
consequences.

EINSTEIN'S approach to the subject seems to me slightly less
abstract than that of GIBBS. This is also confirmed by the fact that
GIBBS made no striving application of his new method, while
EINSTEIN at once proceeded to apply his theorems to a case of utmost
importance, namely to systems of a size suited for demonstrating the
reality of molecules and the correctness of the kinetic theory of
matter.

This was the theory of Brownian movement. EINSTEIN'S papers on
this subject are now easily accessible in a little volume edited and
supplied with notes by R. FORTH, and translated into English by
A. D, GowpER. 3 In the first paper (1905) he sets out to show 'that
according to the molecular-kinetic theory of heat, bodies of micro-
scopically visible size suspended in a liquid will perform movements

8s EINSTEIN'S STATISTICAL THEORIES

of such magnitude that they can be easily observed in a microscope,

on account of the molecular motion of heat,' and he adds that these

movements are possibly identical with the 'Brownian motion' though

his information about the latter is too vague to form a definite

judgment.

The fundamental step taken by EINSTEIN was the idea of raising
the kinetic theory of matter from a possible, plausible, useful
hypothesis to a matter of observation, by pointing out cases where
the molecular motion and its statistical character can be made
visible. It was the first example of a phenomenon of thermal fluctua-
tions, and his method is the classical paradigma for the treatment
of all of them. He regards the movement of the suspended particles
as a process of diffusion under the action of osmotic pressure and
other forces, among which friction due to the viscosity of the liquid
is the most important one. The logical clue to the understanding of
the phenomenon consists in the statement that the actual velocity
of the suspended particle, produced by the impacts of the molecules
of the liquid on it, is unobservable; the visible effect in a finite
interval of time r consists of irregular displacements, the probability
of which satisfies a differential equation of the same type as the
equation of diffusion. The diffusion coefficient is nothing but the
mean square of the displacement divided by ar. In this way EINSTEIN
obtained his celebrated law expressing the mean square displace-
ment for T in terms of measurable quantities (temperature, radius
of the particle, viscosity of the liquid) and of the number of mole-
cules in a gramme-molecule (AVOGADRO'S number JV). By its
simplicity and clarity this paper is a classic of our science.

In the second paper (1906) EINSTEIN refers to the work of SEDEN-
TOPF (Jena) and GOUY (Lyon) who convinced themselves by
observations that the Brownian motion was in fact caused by the
thermal agitation of the molecules of the liquid, and from this
moment on he takes it for granted that the 'irregular motion of
suspended particles' predicted by him is identical with the Brownian
motion. This and the following publications are devoted to the
working out of details (e.g. rotatory Brownian motion) and presenting
the theory in other forms; but they contain nothing essentially new.

I think that these investigations of EINSTEIN have done more than
any other work to convince physicists of the reality of atoms and
molecules, of the kinetic theory of heat, and of the fundamental
part of probability in the natural laws. Reading these papers one
is inclined to believe that at that time the statistical aspect of physics
was preponderant in EINSTEIN'S mind; yet at the same time he
worked on relativity where rigorous causality reigns. His conviction

EINSTEIN'S STATISTICAL THEORIES 83

seems always to have been, and still is to-day, that the ultimate laws
of nature are causal and deterministic, that probability is used to
cover our ignorance if we have to do with numerous particles, and
that only the vastness of this ignorance pushes statistics into the
forefront.

Most physicists do not share this view to-day, and the reason for
this is the development of quantum theory. EINSTEIN'S contribution
to this development is great. His first paper of 1905, mentioned
already, is usually quoted for the interpretation of the photo-
electric effect and similar phenomena (STORES' law of photo-
luminescence, photo-ionisation) in terms of light-quanta (light-darts,
photons). As a matter of fact, the main argument of EINSTEIN is
again of a statistical nature, and the phenomena just mentioned are
used in the end for confirmation. This statistical reasoning is very
characteristic of EINSTEIN, and produces the impression that for him
the laws of probability are central and more important by far
than any other law. He starts with the fundamental difference
between an ideal gas and a cavity filled with radiation: the gas
consists of a finite number of particles, while radiation is described
by a set of functions in space, hence by an infinite number of
variables. This is the root of the difficulty of explaining the law of
black body radiation; the monochromatic density of radiation turns
out to be proportional to the absolute temperature (later known as
the law of RAYLEIGH-JEANS) with a factor independent of frequency,
and therefore the total density becomes infinite. In order to avoid
this, PLANCK (1900) had introduced the hypothesis that radiation
consists of quanta of finite size. EINSTEIN, however, does not use
PLANCK'S radiation law, but the simpler law of WEEN, which is the
limiting case for low radiation density, expecting rightly that here
the corpuscular character of the radiation will be more evident. He
shows how one can obtain the entropy S of black body radiation
from a given radiation law (monochromatic density as function of
frequency) and applies then BOLTZMANN'S fundamental relation
between entropy S and thermodynamic probability W,

where A; is the gas constant per molecule, for determining W '. This
formula was certainly meant by BOLTZMANN to express the physical
quantity S in terms of the combinatory quantity W, obtained by
counting all possible configurations of the atomistic elements of the
statistical ensemble. EINSTEIN inverts this process: he starts from the
known function S in order to obtain an expression for the probability
which can be used as a clue to the interpretation of the statistical

84 EINSTEIN'S STATISTICAL THEORIES

elements. (The same trick has been applied by him later in his work
on fluctuations ; 4 although this is of considerable practical importance,
I shall only mention it, since it introduces no new fundamental
concept apart from that 'inversion'.)

Substituting the entropy derived from WIEN'S law into BOLTZ-
MANN'S formula, EINSTEIN obtains for the probability of finding the
total energy E by chance compressed in a fraction aV of the total
volume V W = a, E *;

that means, the radiation behaves as if it consisted of independent
quanta of energy of size hv and number n E\hv. It is obvious from
the text of the paper that this result had an overwhelming power of
conviction for EINSTEIN, and that it led him to search for confirma-
tion of a more direct kind. This he found in the physical phenomena
mentioned above (e.g. photoelectric effect) whose common feature
is the exchange of energy between an electron and light. The
impression produced on the experimentalists by these discoveries
was very great. For the facts were known to many, but not correlated.
At that time EINSTEIN'S gift for divining such correlations was almost
uncanny. It was based on a thorough knowledge of experimental
facts combined with a profound understanding of the present state
of theory, which enabled him to see at once where something
strange was happening. His work at that period was essentially
empirical in method, though directed to building up a consistent
theory in contrast to his later work when he was more and more
led by philosophical and mathematical ideas.

A second example of the application of this method is the work
on specific heat. 6 It started again with a theoretical consideration
of that type which provided the strongest evidence in EINSTEIN'S
mind, namely on statistics. He remarks that PLANCK'S radiation
formula can be understood by giving up the continuous distribution
of statistical weight in the phase-space which is a consequence of
LIOUVILLE'S theorem of dynamics; instead, for vibrating systems of
the kind used as absorbers and emitters in the theory of radiation
most states have a vanishing statistical weight and only a selected
number (whose energies are multiples of a quantum) have finite
weights.

Now if this is so, the quantum is not a feature of radiation but of
general physical statistics, and should therefore appear in other
phenomena where vibrators are involved. This argument was
obviously the moving force in EINSTEIN'S mind, and it became fertile
by his knowledge of facts and his unfailing judgment of their bearing
on the problem. I wonder whether he knew that there were solid

EINSTEIN'S STATISTICAL THEORIES 85

elements for which the specific heat per mole was lower than its
normal value 5-94 calories, given by the law of DULONG-PETTT, or
whether he first had the theory and then scanned the tables to find
examples. The law of DULONG-PETIT is a direct consequence of the
law of equipartition of classical statistical mechanics, which states
that each co-ordinate or momentum contributing a quadratic term
to the energy should carry the same average energy, namely \ RT
per mole where R is the gas constant; as R is a little less than 2
calories per degree and an oscillator has 3 co-ordinates and 3
momenta, the energy of one mole of a solid element per degree of
temperature should be 6 X %R, or 5-94 calories. If there are
substances for which the experimental value is essentially lower, as
it actually is for carbon (diamond), boron, silicon, one has a
contradiction between facts and classical theory. Another such
contradiction is provided by some substances with poly-atomic
molecules. DRUDE had proved by optical experiments that the
atoms in these molecules were performing oscillations about each
other; hence the number of vibrating units per molecule should be
higher than 6 and therefore the specific heat higher than the
DULONG-PETIT value but that is not always the case. Moreover
EINSTEIN could not help wondering about the contribution of the
electrons to the specific heat. At that time vibrating electrons in
the atom were assumed for explaining the ultra-violet absorption;
they apparently did not contribute to the specific heat, in contradic-
tion to the equipartition law.

All these difficulties were at once swept away by EINSTEIN'S
suggestion that the atomic oscillators do not follow the equipartition
law, but the same law which leads to PLANCK'S radiation formula.
Then the mean energy would not be proportional to the absolute
temperature but decrease more quickly with falling temperature in
a way which still depends on the frequencies of the oscillators. High
frequency oscillators like the electrons would at ordinary tempera-
ture contribute nothing to the specific heat, atoms only if they were
not too light and not too strongly bound. EINSTEIN confirmed that
these conditions were satisfied for the cases of poly-atomic molecules
for which DRUDE had estimated the frequencies, and he showed that
the measurements of the specific heat of diamond agreed fairly well
with his calculation.

But this is not the place to enter into a discussion of the physical
details of EINSTEIN'S discovery. The consequences with regard to the
principles of scientific knowledge were far-reaching. It was now
proved that the quantum effects were not a specific property of
radiation but a general feature of physical systems. The old rule

86 EINSTEIN'S STATISTICAL THEORIES

'natura non facit saltus* was disproved: there are fundamental dis-
continuities, quanta of energy, not only in radiation but in ordinary
matter.

In EINSTEIN'S model of a molecule or a solid these quanta are
still closely connected with the motion of single vibrating particles.
But soon it became clear that a considerable generalization was
necessary. The atoms in molecules and crystals are not independent
but coupled by strong forces. Therefore the motion of an individual
particle is not that of a single harmonic oscillator, but the super-
position of many harmonic vibrations. The carrier of a simple
harmonic motion is nothing material at all ; it is the abstract 'normal
mode', well known from ordinary mechanics. For crystals in particu-
lar each normal mode is a standing wave. The introduction of this
idea opened the way to a quantitative theory of thermodynamics of
molecules and crystals and demonstrated the abstract character of
the new quantum physics which began to emerge from this work. It
became clear that the laws of micro-physics differed fundamentally
from those of matter in bulk. Nobody has done more to elucidate
this than EINSTEIN. I cannot report all his contributions, but shall
confine myself to two outstanding investigations which paved the
way for the new micro-mechanics which physics at large has
accepted to-day while EINSTEIN himself stands aloof, critical,
sceptical, and hoping that this episode may pass by and physics
return to classical principles.

The first of these two investigations has again to do with the law
of radiation and statistics. 6 There are two ways of tackling problems
of statistical equilibrium. The first is a direct one, which one may
call the combinatory method: After having established the weights
of elementary cases one calculates the number of combinations of
these elements which correspond to an observable state; this number
is the statistical probability W, from which all physical properties
can be obtained (e.g. the entropy by BOLTZMANN'S formula). The
second method consists in determining the rates of all competing
elementary processes, which lead to the equilibrium in question.
This is, of course, much more difficult; for it demands not only the
counting of equally probable cases but a real knowledge of the
mechanism involved. But, on the other hand, it carries much further,
providing not only the conditions of equilibrium but also of the
time-rate of processes starting from non-equilibrium configurations.
A classical example of this second method is BOLTZMANN'S and
MAXWELL'S formulation of the kinetic theory of gases; here the
elementary mechanism is given by binary encounters of molecules,
the rate of which is proportional to the number-density of both

EINSTEIN'S STATISTICAL THEORIES 87

partners. From the 'collision equation 9 the distribution function of
the molecules can be determined not only in statistical equilibrium,
but also for the case of motion in bulk, flow of heat, diffusion, etc.
Another example is the law of mass-action in chemistry, established
by GULDBERG and WAAGE; here again the elementary mechanism
is provided by multiple collisions of groups of molecules which
combine, split, or exchange atoms at a rate proportional to the
number-density of the partners. A special case of these elementary
processes is the monatomic reaction, where the molecules of one
type spontaneously explode with a rate proportional to their
number-density. This case has a tremendous importance in nuclear
physics: it is the law of radio-active decay. Whereas in the few
examples of ordinary chemistry, where monatomic reaction has
been observed, a dependence of reaction velocity on the physical
conditions (e.g. temperature) could be assumed or even observed,
this was not the case for radio-activity: the decay constant seemed to
be an invariable property of the nucleus, unchangeable by any
external influences. Each individual nucleus explodes at an un-
predictable moment; yet if a great number of nuclei are observed,
the average rate of disintegration is proportional to the total number
present. It looks as if the law of causality is put out of action for
these processes.

Now what EINSTEIN did was to show that PLANCK'S law of radia-
tion can just be reduced to processes of a similar type, of a more or
less non-causal character. Consider two stationary states of an atom,
say the lowest state i and an excited state a. EINSTEIN assumes that
if an atom is found to be in the state 2 it has a certain probability of
returning to the ground state i, emitting a photon of a frequency
which, according to the quantum law, corresponds to the energy
difference between the two states; i.e. in a big assembly of such
atoms the number of atoms in state 2 returning to the ground state i
per unit time is proportional to their initial number exactly as
for radio-active disintegration. The radiation, on the other hand,
produces a certain probability for the reverse process i -> 2 which
represents absorption of a photon of frequency v 12 and is propor-
tional to the radiation density for the frequency.

Now these two processes alone balancing one another would not
lead to PLANCK'S formula; EINSTEIN is compelled to introduce a
third one, namely an influence of the radiation on the emission
process 2 i, 'induced emission,' which again has a probability
proportional to the radiation density for v ia .

This extremely simple argument together with the most elementary
principle of BOLTZMANN'S statistics leads at once to PLANCK'S

Qg EINSTEIN*S STATISTICAL THEORIES

formula without any specification of the magnitude of the transition
probabilities. EINSTEIN has connected it with a consideration of the
transfer of momentum between atom and radiation, showing that
the mechanism proposed by him is not consistent with the classical
idea of spherical waves but only with a dart-like behaviour of the
quanta. Here we are not concerned with this side of EINSTEIN'S
work, but with its bearing on his attitude to the fundamental
question of causal and statistical laws in physics. From this point of
view this paper is of particular interest. For it meant a decisive step
in the direction of non-causal, indeterministic reasoning. Of course,
I am sure that EINSTEIN himself was and is still convinced that
there are structural properties in the excited atom which determine
the exact moment of emission, and that probability is called in only
because of our incomplete knowledge of the pre-history of the atom.
Yet the fact remains that he has initiated the spreading of indeter-
ministic statistical reasoning from its original source, radio-activity,
into other domains of physics.

Still another feature of EINSTEIN'S work must be mentioned which
was also of considerable assistance to the formulation of indeter-
ministic physics in quantum mechanics. It is the fact that it follows
from the validity of PLANCK'S law of radiation that the probabilities
of absorption (i -> 2) and induced emission (2 -> i) are equal. This
was the first indication that interaction of atomic systems always
involves two states in a symmetrical way. In classical mechanics an
external agent like radiation acts on one definite state, and the
result of the action can be calculated from the properties of this
state and the external agent. In quantum mechanics each process is
a transition between two states which enter symmetrically into the
laws of interaction with an external agent. This symmetrical
property was one of the deciding clues which led to the formulation
of matrix mechanics, the earliest form of modern quantum mechan-
ics. The first indication of this symmetry was provided by EINSTEIN'S
discovery of the equality of up- and down-ward transition proba-
bilities.

The last of EINSTEIN'S investigations which I wish to discuss in
this report is his work on the quantum theory of monatomic ideal
gases. 7 In this case the original idea was not his but came from an
Indian physicist, S. N. BOSE; his paper appeared in a translation by
EmsTEDsr 8 himself who added a remark that he regarded this work
as an important progress. The essential point in Boss's procedure is
that he treats photons like particles of a gas with the method of
statistical mechanics but with the difference that these particles are
not distinguishable. He does not distribute individual particles over

EINSTEIN'S STATISTICAL THEORIES 89

a set of states, but counts the number of states which contain a given
number of particles. This combinatory process together with the
physical conditions (given number of states and total energy) leads
at once to PLANCK'S radiation law. EINSTEIN added to this idea the
suggestion that the same process ought to be applied to material
atoms in order to obtain the quantum theory of a monatomic gas.
The deviation from the ordinary gas laws derived from this theory
is called 'gas degeneracy.' EINSTEIN'S papers appeared just a year
before the discovery of quantum mechanics ; one of them contains
moreover (p. 9 of the second paper) a reference to DE BROGLIE'S
celebrated thesis, and the remark that a scalar wave field can be
associated with a gas. These papers of DE BROGLIE and EINSTEIN
stimulated SCHRODINGER to develop his wave mechanics, as he
himself confessed at the end of his famous paper. 9 It was the same
remark of EINSTEIN'S which a year or two later formed the link
between DE BROGLIE'S theory and the experimental discovery of
electron diffraction; for, when DAVTSSON sent me his results on the
strange maxima found in the reflexion of electrons by crystals, I
remembered EINSTEIN'S hint and directed ELSASSER to investigate
whether those maxima could be interpreted as interference fringes
of DE BROGLIE waves. EINSTEIN is therefore clearly involved in the
foundation of wave mechanics, and no alibi can disprove it.

I cannot see how the BOSE-EINSTEIN counting of equally probable
cases can be justified without the conceptions of quantum mechanics.
There a state of equal particles is described not by noting their
individual positions and momenta, but by a symmetric wave function
containing the co-ordinates as arguments; this represents clearly
only one state and has to be counted once. A group of equal particles
even if they are perfectly alike, can still be distributed between two
boxes in many ways you may not be able to distinguish th.em
individually but that does not affect their being individuals. Al-
though arguments of this kind are more metaphysical than physical,
the use of a symmetric wave function as representation of a state
seems to me preferable. This way of thinking has, moreover, led to
the other case of gas degeneracy, discovered by FERMI and DIRAG,
where the wave function is skew, and to a host of physical conse-
quences confirmed by experiment.

The BOSE-EINSTEIN statistics was, to my knowledge, EINSTEIN'S
last decisive positive contribution to physical statistics. His following
work in this line, though of great importance by stimulating thought
and discussion, was essentially critical. He refused to acknowledge
the claim of quantum mechanics to have reconciled the particle
and wave aspects of radiation. This claim is based on a complete

go EINSTEIN'S STATISTICAL THEORIES

re-orientation of physical principles : causal laws are replaced by
statistical ones, determinism by indeterminism. I have tried to show
that EINSTEIN himself has paved the way for this attitude. Yet some
principle of his philosophy forbids him to follow it to the end. What
is this principle?

EINSTEIN'S philosophy is not a system which you can read in a
book; you have to take the trouble to abstract it from his papers on
physics and from a few more general articles and pamphlets. I have
found no definite statement of his about the question 'What is
Probability?'; nor has he taken part in the discussions going on
about VON MISES' definition and other such endeavours. I suppose
he would have dismissed them as metaphysical speculation, or even
joked about them. From the beginning he has used probability as
a tool for dealing with nature just like any scientific device. He has
certainly very strong convictions about the value of these tools. His
attitude toward philosophy and epistemology is well described in
his obituary article on ERNST MAGH: IQ

Nobody who devotes himself to science from other reasons than super-
ficial ones, like ambition, money making, or the pleasure of brain-sport,
can neglect the questions, what are the aims of science, how far are its
general results true, what is essential and what based on accidental features
of the development?

Later in the same article he formulates his empirical creed in these
words:

Concepts which have been proved to be useful in ordering things easily
acquire such an authority over us that we forget their human origin and
accept them as invariable. Then they become 'necessities of thought,'
'given a priori? etc. The path of scientific progress is then, by such errors,
barred for a long time. It is therefore no useless game if we are insisting
on analysing current notions and pointing out on what conditions their
justification and usefulness depends, especially how they have grown from
the data of experience. In this way their exaggerated authority is broken.
They are removed, if they cannot properly legitimate themselves; corrected,
if their correspondence to the given things was too negligently established;
replaced by others, if a new system can be developed that we prefer for
good reasons.

That is the core of the young EINSTEIN, thirty years ago. I am sure
the principles of probability were then for him of the same kind as
all other concepts used for describing nature, so impressively
formulated in the lines above. The EINSTEIN of to-day is changed. I
translate here a passage of a letter from him which I received about
four years ago (November yth, 1944) : In our scientific expectation
we have grown antipodes. You believe in God playing dice and I
in perfect laws in the world of things existing as real objects, which
I try to grasp in a wildly speculative way.' These speculations

EINSTEIN'S srAtisncAL THEORIES gi

distinguish indeed his present work from his earlier writings. But if
any man has the right to speculate it is he whose fundamental results
stand like a rock. What he is aiming at is a general field-theory which
preserves the rigid causality of classical physics and restricts proba-
bility to masking our ignorance of the initial conditions or, if you
prefer, of the pre-history, of all details of the system considered.
This is not the place to argue about the possibility of achieving
this. Yet I wish to make one remark, using EINSTEIN'S own picturesque
language : If God has made the world a perfect mechanism, he has
at least conceded so much to our imperfect intellect that, in order
to predict little parts of it, we need not solve innumerable differential
equations but can use dice with fair success. That this is so I have
learned, with many of my contemporaries, from EINSTEIN himself.
I think, this situation has not changed much by the introduction of
quantum statistics; it is still we mortals who are playing dice for
our little purposes of prognosis God's actions are as mysterious in
classical Brownian motion as in radio-activity and quantum radia-
tion, or in life at large.

EINSTEIN'S dislike of modern physics has not only been expressed
in general terms, which can be answered in a similarly general and
vague way, but also in very substantial papers in which he has
formulated objections against definite statements of wave mechanics.
The best known paper of this kind is one published in collaboration
with PODOLSKY and RosEN. 11 That it goes very deep into the logical
foundations of quantum mechanics is apparent from the reactions it
has evoked. NIELS BOHR has answered in detail; SCHRODINGER has
published his own sceptical views on the interpretation of quantum
mechanics; REICHENBACH deals with this problem in the last
chapter of his excellent book. Philosophic Foundations of Quantum
Mechanics, and shows that a complete treatment of the difficulties
pointed out by EINSTEIN, PODOLSKY, and ROSEN needs an overhaul
of logic itself. He introduces a three-valued logic, in which apart
from the truth-values c true s and 'false', there is an intermediate one,
called 'indeterminate', or, in other words, he rejects the old principle
of 'tertium non datur\ as has been proposed long before, from purely
mathematical reasons, by BROUWER and other mathematicians. I
am not a logician, and in such disputes always trust that expert
who last talked to me. My attitude to statistics in quantum mechanics
is hardly affected by formal logic, and I venture to say that the
same holds for EINSTEIN. That his opinion in this matter differs
from mine is regrettable, but it is no object of logical dispute between
us. It is based on different experience in our work and life. But in
spite of this, he remains my beloved master.

92 EINSTEIN'S STATISTICAL THEORIES

REFERENCES

A. EINSTEIN, Annalen der Physik (1902) (4), 9, p. 477.
A. EINSTEIN, Annalen der Physik (1903) (4), n, p. 170.
A. EINSTEIN, Investigations on the Theory of the Brownian Movement. London:

Methuen & Go. (1926).

A. EINSTEIN, Annalen der Physik (1906) (4), 19, p. 373.
A. EINSTEIN, Annalen der Physik (1907) (4), 22, p. 180.
A. EINSTEIN, Phys. . (1917), 18, p. 121.
A. EINSTEIN, BerL Ber. (1924) p. 261; (1925) p. 318.
S. N. BOSE (1924) Zeitschriftfur Physik, 26, 178.
E. SCHRODINGER, Annalen der Physik (1926) (4), 70, p. 361; s. p. 373.

10 A. EINSTEIN, Phys. . (1916) 17, p. 101.

11 A. EINSTEIN, B. PODOLSKY and N. ROSEN (1935) Phys. Rev., 47, 777.

PHYSICS AND METAPHYSICS

[Joule Memorial Lecture, 1950. First published in Vol. 91 of Memoirs and
Proceedings of the Manchester Literary and Philosophical Society, 1049-50.]

HpHE subject which I have chosen to commemorate the great
* discoverer of the first law of thermodynamics has nothing to do
with JOULE'S own work. In fact I would be quite incompetent to
deal with experiments, and my knowledge of the history of JOULE'S
discovery in connection with the work of his contemporaries,
ROBERT MAYER and HELMHOLTZ, is second-hand. I propose to
speak about a very general matter. It is on the borderline of two
fields of research, and this seems to imply that I am familiar with
both of them. However, though I feel on fairly stable ground when
speaking about physics, I cannot claim in any way to be expert in
what is customarily treated in philosophical books and lectures under
the title of metaphysics. What I know of it is more or less the recollec-
tion from my student days, refreshed by some sporadic reading.
Long years of neglect have not deleted the deep impression received
in my youth by the age-old attempts to answer the most urgent
questions of the human mind: the questions about the ultimate
meaning of existence, about the Universe at large and our part in it,
about life and death, truth and error, goodness and vice, God and
eternity. But just as deep as this impression of the importance of the
problems is the memory of the futility of the endeavour. There
seemed to be no steady progress as we find in the special sciences,
and like so many others, I turned my back to philosophy and found
satisfaction in a restricted field where problems can actually be
solved. Yet getting old, I feel, again like many others, whose pro-
ductive powers are declining, the desire to summarise the results of
the scientific search in which, during several decades, I have taken
a small part, and that leads unavoidably back to those eternal
questions which go under the title of metaphysics.

Let me quote two definitions of metaphysics by modern philoso-
phers. WTT.T.TAM JAMES says: 'Metaphysics is an unusually stubborn
effort to think dearly. 3 BERTRAND RUSSELL says: 'Metaphysics, or
the attempt to conceive the world as a whole by means of thought/

These formulations stress two important aspects; one the method:
stubborn clear thinking; the other the object: the world as a whole.
But is every case of stubborn clear thinking metaphysics? Every
scientist, every historian, philologist, even theologian would claim

93

94 PHYSICS AND METAPHYSICS

to think clearly. On the other hand, the world as a whole is a subject
not only vast, but definitely not closed, open to new discovery at any
moment, therefore not exhausted and probably inexhaustible; in
short, the world known to us is never a whole. I shall return to this
point at the end.

I propose to use the word metaphysics in a more modest way,
regard to method and subject as well, namely as an investigation into
the general features of the structure of the world and our methods
to deal with this structure. I wish to discuss in particular the question
whether the progress of physics has contributed anything essential
to this problem. This progress of physics has been, as we are all
aware, somewhat sensational during the last few years, and the aspect
of the physical world has thoroughly changed in the half century of
my own scientific life. Yet the methods of the physicist have always
remained essentially the same : experimenting, observing regularities,
formulating mathematical laws, predicting new phenomena with
the help of these laws, combining the different empirical laws in
coherent theories, which satisfy our sense of harmony and logical
beauty, and testing these theories again by prediction. These succes-
ful predictions are the highlights of theoretical physics, as we have
witnessed in our day in the case of DE BROGUE'S waves, of DIRAC'S
positron, YUKAWA'S meson and many more such cases.

The power of prediction is the main claim of physics. It is based
on the acceptance of the principle of causality, which, in its most
general form, means the assumption of invariable laws of Nature.
Yet you will all have heard that modern physics has been led to
doubt this principle. Here is the first metaphysical conception on
which I wish to make some comments.

Closely connected with it is the conception of reality. The sceptical
attitude in regard to causality has arisen in atomic physics where the
objects are not immediately accessible to our senses but only in-
directly, with the help of more or less complicated apparatus. These
ultimate objects of physics are particles, forces, fields, etc. ; what
kind of reality can one ascribe to them? This leads to the more
general question of the relation between subject and object, of the
existence of an objective physical world independent of the observing
subject, and thus back to RUSSELL'S problem, whether a conception
of the world as a whole is actually possible.

The cause-effect relation is used in ordinary life in two rather
different ways, which may be illustrated by the following two
statements:

'The capitalistic system is the cause of economic crises', and 'the
economic crisis of 1930 was caused by a panic at the New York

PHYSICS AND METAPHYSICS 95

exchange'. One states a general rule or law, independent of time;
the other declares one definite event to be the necessary sequence of
another definite event. Both cases have the idea of necessity in
common, a conception of a somewhat mysterious character which I
feel completely unable to analyse further, and which I am willing
to accept as metaphysical. Classical physics has officially adopted
the second form of causality, as a necessary sequence in time. This
came about through the discovery of the fundamental laws of
mechanics by GALILEO and NEWTON, laws which allow the predic-
tion of future events from previous ones or vice versa. In other
words, these laws are deterministic: a world governed solely by
them would be a gigantic machine; the complete knowledge of the
situation at a given time would determine the situation at any other
time. This kind of determinism was regarded by the physicists of the
last century as the only rational interpretation of causality, and by
using it they boasted that they had eliminated from physics the last
remnants of metaphysical thinking.

Now it seems to me that this identification of determinism and
causality is quite arbitrary and confusing. There are deterministic
relations which are not causal; for instance, any time table or
programmatic statement.

To take an absurdly obvious case, you could predict from the
programme of a pantomime the sequence of the scenes but would
hardly say that the acrobats of scene No. 5 had caused the love
scene No. 6. To return to science. The Ptolemaic system of the
Cosmos is a deterministic but not a causal interpretation, and the
same can be said about COPERNICUS' cycles and KEPLER'S ellipses.
They are all, in the usual scientific terminology, kinematic descrip-
tions, but not causal explanations. For no cause of the phenomena
is given except the ultimate cause of the creator's will. Then came
the dynamical theories of GALILEO and NEWTON. If one sticks to the
programme that the only aim of a theory is deterministic prediction,
the progress made by the introduction of dynamics into astronomy
can merely be seen in a considerable condensation and simplifica-
tion of the laws. When I was a student in Germany, fifty years ago,
this standpoint, skilfully formulated by KIRGHHOFF, was dominant
and is still widely shared.

I think that the discovery of mechanics was a much more funda-
mental affair. GALILEO showed that a certain quantity, connected
with the motion of a body, namely its acceleration, is independent
of the body and of its motion and only dependent on its position
relative to the earth; and NEWTON showed the same for the planets
where the acceleration depends only on the distance from the sun.

g6 PHYSICS AND METAPHYSICS

This appears to me something more than a short and efficient
description of facts. It means the introducing of a quantitative
expression of the cause-effect relation in its most general form
through the concept of force. It introduces the idea, foreign to the
older kinematic theories, that one set of data (here positions)
'causes' another set of data (here accelerations). The word 'causes'
means just 'determines quantitatively', and the law offeree expresses
in detail how the effect depends on the cause.

This interpretation of the laws of mechanics brings them into
line with the ordinary practice of the scientist. An experiment is
planned, i.e. certain conditions of observation are produced; then
the effect is observed, sometimes at a future date, but more often
all the time while the conditions hold. It is the timeless relation
between observation and conditions of observation (apparatus)
which is the real object of science. I suggest that this is the actual
meaning of the principle of causality, as distinct from determinism,
which is a special, and almost accidental property of the mechanical
laws (due to the fact that one kind of the quantities involved are
accelerations, i.e. time derivatives).

If one looks on the history of physics during the last centuries
from this point of view (as I have tried in my Waynflete Lectures,
which have recently been published under the title 'Natural
Philosophy of Cause and Chance') one gets the following im-
pression:

Physics has used just this timeless cause-effect relation in its
everyday practice but another notion in the theoretical interpreta-
tion. There causality was taken as synonymous with determinism,
and as the deterministic form of the mechanical laws is an empirical
fact, this interpretation was hailed as a great achievement in
eliminating dark metaphysical concepts. However, these concepts
have a strange way of asserting themselves. Causality has in every-
day life two attributes, which for shortness I shall call the principles
of contiguity and of antecedence. The first states that things can act
only on neighbouring things, or through a chain of things in con-
tact, and the second that if cause and effect refer to situations at
different times the cause should be prior to the effect.

Both principles are violated by Newtonian mechanics, as the
gravitational force acts over any distance of empty space, and as
the laws of motion connect two configurations at different times in a
perfectly symmetric and reversible way. One can regard the whole
development of classical physics as a struggle to re-establish these
two essential features of the concepts of cause and effect. The
methods to preserve contiguity were mathematically developed, by

PHYSICS AND METAPHYSICS 97

CAUCHY and others, by extending mechanics to continuous media;
the idea of contiguity played a leading part in FARADAY'S researches
on electricity and magnetism, and led to MAXWELL'S concept of a
field offeree propagating itself with finite velocity, which was soon
confirmed by HERTZ'S discovery of electro-magnetic waves. Finally
NEWTON'S case was brought into line with contiguity through
EINSTEIN'S relativistic theory of the gravitational field. No modern
theory of interaction is thinkable which violates this principle.

Antecedence has a much more tortuous history and not a happy
end. It took much effort to discover that in physics the distinction
between past and future was linked with the irreversibility of heat
phenomena here we remember JOULE as one of the central figures
and to reconcile this result with the reversibility of mechanics
through the development of atomistics and statistical methods. I
think that this work, initiated by MAXWELL, BOLTZMANN, GIBBS
and EINSTEIN, is one of the greatest achievements in science. The
deterministic interpretation of causality could be maintained for
the atomic world and yet the apparent validity of antecedence
understood as an effect of the statistical law of large numbers.
However, this interpretation carried the germ of self-destruction of
one of its pillars: it opened the way to the study of the atomic world,
and the result was that the presupposed validity of Newtonian
mechanics in this microscopic world was wrong. The new quantum
mechanics does not allow a deterministic interpretation, and since
classical physics has identified causality with determinism, the doom
of the causal explanation of nature seems to have come.

I am much opposed to this view. It does not matter much in
discussions between scientists who know exactly what they are
talking about; but it is harmful if used in describing the last results
of science to the non-scientific world. Extremes are always harmful.
The deterministic mechanistic view produced a philosophy which
shut its eyes against the most obvious facts of experience; but a
philosophy which rejects not only determinism, but causation,
altogether seems to me just as absurd. I think that there exists a
reasonable definition of the cause-effect relation which I have
already mentioned: that a certain situation depends on another one
(irrespective of time) in a way describable by quantitative laws.

I shall indicate how this is still true in quantum mechanics in
spite of its indeterministic character, and how the apparent loss is
compensated by another fundamental principle, called complement-
arity, which will be of great philosophical and practical importance.

This new conception is due to NIELS BOHR, the great Danish
physicist, who was one of the leaders in the development of quantum

98 PHYSICS AND METAPHYSICS

mechanics, not only in regard to physics itself, but also to the
philosophical implications. I was fortunate enough to listen to his
Gifford Lectures, given in Edinburgh last autumn, which I hope
will be published in the not too far distant future. I cannot give you
an account of his ideas in the short time left to me, but only try to
outline the main points and to bring them into line with my slightly
different formulations.

As you will know, PLANCK'S fundamental law of quantum
theory connects an energy E with a frequency v, by the simple
formula E = hv 9 where h is a constant. This was later extended by
EINSTEIN and DE BROGUE, from the number of vibrations v per unit
time to the number of waves K per unit length, which is connected
with a mechanical momentum p by the corresponding formula
p = kK> with the same constant h.

That this is so, has been confirmed by innumerable direct ex-
periments and more or less indirect inferences from observations.
Whenever a process can be resolved in periodic components with
definite periods in time and space, i.e. with definite v and /c, the
effect of it on the motion of particles consists in transferring energy
and momentum according to this law. This empirical fact must be
accepted as undeniable before its implications can be discussed.

Now this fact is so extremely strange that it took many years
before physicists began to consider it seriously, and NIELS BOHR
himself has used the word 'irrational' to describe the new feature of
the physical world discovered by PLANCK. Why irrational ? Because
energy and momentum of a particle are, by their definition, related
to an extremely small region of space, practically to a point, while
frequency and wave number, also by their definition, are related to
a very large, theoretically infinite, extension of space and time. This
latter point will perhaps not appear so obvious as the first; you may
say, I hear a tone of a piano string well defined even if it is played
extremely staccato. This is practically true, because our ear is not
a very sensitive instrument to discover tiny distortions. But the
telecommunication engineer is familiar with the fact that there is a
distortion. A tone lasting only a short time, comparable with the
period, is not pure any more but accompanied by other tones, with
frequencies spread out over a little interval Ay around the original
one; and if the duration is getting shorter and shorter this interval
becomes larger and larger, until no tone is heard but a noise, a
crack. As modern telecommunication is based on the principle of
modulation, i.e. of interrupting a high frequency current in the
rhythm of signals or modifying its strength according to the relatively
slow vibrations of speech or music, it is obvious that there is a limit

PHYSICS AND METAPHYSICS 99

to the perfection of transmission : if A/ is the duration of a tone of
frequency v, there is a relative limit of recognisability given in
order of magnitude by A/ . Av < ' i. An excellent account of these
problems has been given by Dr. GABOR in this country, and in
America a book with the title "Cybernetics", i.e., the science of
governing namely by sending out signals and orders has been
published by NORBERT WIENER, which, though full of rather abstruse
mathematics, has made quite a sensational stir. In fact the mathe-
matical analysis of these relations, which has its roots, almost one
and a half centuries ago, in an investigation by FOURIER on the
conduction of heat, is rather simple. The main point is that the ideal,
or pure, or harmonic vibration to which alone a sharp frequency can
be ascribed, appears in a time-amplitude diagram as an endless
train of sinoidal waves. Every other curve, for instance a wave
restricted to a finite interval of time, is a superposition of harmonic
waves and has a whole Spectrum' of v-values. The same holds for
real waves expanding hi space where apart from the periodicity in
time one has a periodicity in space measured by the wave number
/c; between the length AZ of a train of waves and the width A/c
of the /c-spectrum one has the relation

A/ . A/c~ i.

There is no other logical way of dealing with periodic processes
or waves than this Fourier analysis, and practical applications have
amply confirmed the theory.

Let us return to quantum physics. The 'irrationality' can now be
formulated more precisely; in order to define v and K sharply, one
has to have very small Av and A/c, hence a very long duration
A^i/Av and spatial extension A/<-^i/A/c. So far nothing is
different from the case of telecommunication, and nothing para-
doxical. But if one uses the relations E = hv 9 p = fiK and re-writes
the limiting relations in the form

A* . AE ~ h, AZ . Aj& ~ h,

they indicate a paradoxical situation: that with a tiny particle of
sharp energy and momentum (i.e. small AJ? and Ap) there are
associated long intervals of time and space A* and A/. What can the
meaning of A and AZ be?

The only possible answer is, that they mean the limits for deter-
mining the position of the particle in time and space. They are
indeed nothing but HEISENBERG'S much discussed uncertainty
relations.

IOO PHYSICS AND METAPHYSICS

Thus it is seen that the very first quantum laws lead necessarily
to a mutual restriction in the accuracy attainable in space-time
location on the one hand, energy-momentum determination on the
other. As BOHR has stressed again and again, we are confronted here
with a logical alternative : either to deny the validity of an enormous
amount of experience confirming the quantum laws E hv,
p = fiK, or to accept the existence of those limits for the determina-
tion of such pairs of quantities, as time-energy, co-ordinate-momen-
tum, which in the mechanical terminology are called conjugate.
The most remarkable thing is that in spite of the completely new
and revolutionary basic situation it was possible to develop a quan-
tum mechanics which is a straightforward generalization of classical
mechanics, extremely similar in mathematical form and consider-
ably more perfect in its structure. It is true that the simple way of
describing variable quantities as functions of time has to be given
up and a more abstract method introduced where physical quan-
tities are represented by non-commuting symbols (i.e. symbols
with which one can form sums and products; the value of the latter
however depends on the order of the factors).

I shall never forget the thrill which I experienced when I suc-
ceeded in condensing HEISENBERG'S ideas on quantum conditions
in the mysterious equation pq qp = A/2m, which is the centre
of the new mechanics and was later found to imply the uncertainty
relations.

The transition from the symbols to actual quantities which can
be measured is made by the introduction of a quantity called wave
function, which describes the state in which a system is found as far
as it can be described : its square is the probability density for finding
the given data (e.g. co-ordinates of particles) in a given small
region, analogous to the distribution function of ordinary statistics.
There is, however, a fundamental difference.

Suppose two beams of particles coming from the same source
counted separately, give the results fa* and fa 2 ; if by a suitable
arrangement they can be made to overlap and be counted together,
the result is (fa + fa) 2 , which differs from the sum fa* + fa*
(by 2 fa fa). One has 'interference' of probabilities, as is well known
from the case of light quanta or photons, the particles whose
abundance is measured by the square of the intensity of an electro-
magnetic wave. But I cannot enter into a technical description of
wave mechanics which has been developed from the foundations
laid by DE BROGLIE, through the ingenuity of SCHRODINGER, DIRAG,
and others. It suffices to say that a wave function ft can be regarded
as a packet of harmonic waves of different v and /c, and that the

PHYSICS AND METAPHYSICS id I

physical quantities like co-ordinates, momenta, energies q, p, E,
are operators distorting the ^-function, and thus determining the
strengths of the harmonic components of the packet from which,
by squaring, the probability of the appearance of particles with
given E = hv, p = HK is obtained.

Thus the new mechanics is essentially statistical and, in regard to
the distribution of particles, completely ^deterministic. Yet it
preserves, strangely enough, some similarity to classical mechanics,
as the law of propagation of the function i/r, the so-called SCHROD-
INGER equation, is of the same type as the wave equations of elasticity
or electro-magnetism. One has therefore the somewhat paradoxical
situation, that there is no determinism for physical objects, like
small particles, but for the probability of their appearance. Yet
this determination of the ^-function needs extremely much more
data than we are accustomed to in classical mechanics (initial
positions and velocities of particles). In fact it needs a knowledge,
or at least a hypothetical knowledge, of ^ everywhere at given time
and at the boundaries at all times, for the region and period in
question; or in other words: predictions even of probabilities alone
can be made only with reference to the whole situation, to the
apparatus used. One must decide beforehand which feature one
wishes to investigate, and one must construct the instrument corres-
pondingly. Then the effect can be predicted, in terms of particles,
as a probability of their appearance under the conditions of the
experiment (e.g. with given momentum), either at a certain finite
region independent of time, or at a later time. That is in complete
harmony with the meaning of causality which I have suggested.
The use of this terminology is not a mere decoration; for it is
essential to be clear that here the metaphysical, irreducible concept
of necessity in the relation of two sets of things is postulated, which
is the characteristic feature of the scientific attitude to the world.

Summarizing, we may say that while classical physics assumes
natural phenomena going on, independent of the incidence of obser-
vation and describable without reference to observation, quantum
physics claims only to describe and predict a phenomenon in
relation to a well defined mode of observation or instrumental
arrangement. But one can, of course, use different instruments for
observing the same class of phenomena; the propagation of light for
instance can be investigated by prisms or gratings with help of
photographic plates or Geiger counters. If every arrangement, from
the standpoint of quantum mechanics, has to be considered separ-
ately, what is the common feature of all of them? For instance, if
by one arrangement we can determine the spatial distribution of

102 PHYSICS ANB METAPHYSICS

electrons, by others their distribution in energy, how can we know
if and when we have exhausted all possibilities?

This question has been discussed in detail by NIELS BOHR under
the title Complementarity. It is true that he presents his ideas in a
little different way: he is keenly intent to show by simple examples
how one can intuitively understand the wholeness of an experi-
mental situation and the mutual exclusiveness and complementarity
of two such situations by using nothing but the uncertainty principle
in its simplest form. I think that his motive in spending much
ingenuity and effort on this task is the tragic situation that the
philosophical attitude accepted by him and presented here by
myself, also accepted by the whole international community of
atomic physicists, has not found favour in the eyes of just those men
who have contributed most to the development of quantum theory,
PLANCK and EINSTEIN. PLANCK preserved always a cautious attitude
to the revolutionary consequence of his own discovery, but Einstein
went further and made repeated efforts to show by simple examples
that the renunciation of determinism and the uncertainty relation
are wrong. Just these examples have been studied by BOHR, in
collaboration with Professor ROSENFELD, who is now here in Man-
chester; in every case EINSTEIN'S objections could be refuted by a
refined study of the experimental situation. The main point is that
an instrument, by its very definition, is a physical system whose
structure can be described in ordinary language and whose function-
ing in terms of classical mechanics. Indeed, this is the only way in
which we can communicate about it with one another. For instance
any spatial location needs a rigid frame, any measurement of time a
mechanical clockwork, while on the other hand a determination of
momentum and energy needs a break of rigidity and mechanical
connection, a freely movable part of the instrument to which the laws
of conservation can be applied. Now BOHR shows that these two
types of arrangement are mutually exclusive and .complementary,
in exact agreement with the results of the theory. If you use a
diaphragm with a slit for fixing a co-ordinate of a particle passing
through it, the diaphragm must be fixed to the frame of the instru-
ment; if you wish to know whether a particle has really passed .the
slit, the diaphragm must be movable so as to be able to recoil. You
cannot have it both ways. By taking this complementarity into
account, one can describe experiments without contradictions.
Sometimes this is not quite easy. I cannot refrain from indicating
one example, which EINSTEIN brought forward at the Solvay, Con-
ference in 1930, with the purpose of showing that it was possible .to
determine the exact time of an atomic -event and the change- of

PHYSICS AND METAPHYSICS 103

energy simultaneously, namely by making use of the relation
E = me 2 , derived from the theory of relativity. One has only to
determine the mass m by weighing, to find the energy E. Assume
radiation enclosed in a box with a shutter which is worked by a
clockwork inside the box and allows the escape of a given amount of
energy, one or several photons, at a moment fixed with any accuracy
desired. Moreover, you could weigh the whole box before and after
this event and thus measure the energy released with any accuracy
wanted, in contradiction to the reciprocal indeterminacy of time
and energy assumed by quantum mechanics. This seems to be a
serious challenge. BOHR'S answer is that the emission of energy is
equivalent to a change of weight and therefore a displacement of
the balance which must be compensated. But this displacement in
the gravitational field of the earth is coupled with a change of rate
of the clock. All these effects can be determined within limits of
accuracy which depend on one another and produce the result,
that EINSTEIN'S method does not work.

I shall now describe this in more detail. As the uncertainty AT
of a measurement of time is proportional to the time measured, we
must avoid delay and hang our box directly on the balance. If the
shutter is opened the balance will move and can then be readjusted
with an accuracy A#. As this happens in the field of the earth g,
there is a change of gravitational potential O = gq, at the place of
the clock, which is fixed with a latitude AO = g&q. The reading
of the clock in the time interval T necessary for this according to
the general theory of relativity, will have a relative uncertainty

T c* c* '

If in this time Tthe weight of the box is determined with an accuracy
Am, one has, from NEWTON'S law of motion, for the latitude in
measuring the momentum of the box, Ap = g Am T. By sub-
stituting the values of A# and Ap from these relations in A/> . Ag

rw h, one finds

t& \T
A~A/>. A = Amr-^ = c 2 AmAr= AE. AT,

according to the relativistic connection between mass and energy.
Hence it is impossible to determine energy and time of release as
well, with arbitrary accuracy.

You will find numerous examples in BOHR'S GifFord Lectures.
While I wrote this, a new book came into my hands, Albert Einstein,
Philosopher and Scientist (The Library of Living Philosophers: Editor,
Paul Arthur Schilpp, 1949), which contains articles of many philo-

104 PHYSICS AND METAPHYSICS

sophers and theoretical physicists on different aspects of EINSTEIN'S
work, amongst them also one by NIELS BOHR and one by myself.
The most interesting part of the work is a scientific autobiography
by EINSTEIN, and a summarizing article in which he answers the
criticism in the previous essays. This is most fascinating reading,
but with all respect to the great physicist, I cannot accept his
arguments against the philosophy of the quantum physicists. All
essential points are treated in BOHR'S article where he gives a
delightful account of a number of discussions he had with EINSTEIN.
But the latter persists in his opposition, and declares himself firmly
convinced that the present theory, though logically consistent, is an
incomplete description of physical systems. His main arguments
are not so much derived from considerations of causality, but from
the new attitude to the meaning of physical reality which it implies.
Let me quote his words (p. 672) : Tor me ... the expectation that
the adequate formulation of the universal laws involves the use of all
conceptual elements which are necessary for a complete description,
is more natural*, namely than the ideas of the quantum physicists,
and he insists that the emission of, say, an a-particle by a radioactive
atom with definite energy must happen at a definite time predictable
from theory otherwise he calls the description conceptionally
incomplete. Yet he, himself, has taught us in the case of relativity
that this argument is wrong. There you have an infinite number of
equivalent inertial systems, each of which can be assumed to be at
rest with the same right. But there is no way of deciding experi-
mentally which is truly or absolutely at rest. EINSTEIN'S opponents
pointed out that they regarded a description of the world as con-
ceptionally incomplete which denied the existence of a system
absolutely at rest, even if there is no experimental way of finding it.
This antirelativistic argument is just as strong as EINSTEIN'S anti-
quantistic one, as everybody has experienced who was asked to
conceive a light wave without a material ether as a carrier of the
vibrations.

The generation to which EINSTEIN, BOHR and I belong, was
taught that there exists an objective physical world, which unfolds
itself according to immutable laws independent of us; we are watch-
ing this process as the audience watches a play in a theatre.
EINSTEIN still believes that this should be the relation between the
scientific observer and his subject. Quantum mechanics, however,
interprets the experience gained in atomic physics in a different
way. We may compare the observer of a physical phenomenon
not with the audience of a theatrical performance, but with that of
a football game where the act of watching, accompanied by

PHYSIOS AND METAPHYSICS 1(>5

applauding or hissing, has a marked influence on the speed and
concentration of the players, and thus on what is watched. In fact,
a better simile is life itself, where audience and actors are the same
persons. It is the action of the experimentalist who designs the
apparatus, which determines essential features of the observations.
Hence there is no objectively existing situation, as was supposed to
exist in classical physics. Not only EINSTEIN, but also others who
are opposed to our interpretation of quantum mechanics, have said
that under these circumstances there is no objectively existing
external world, no sharp distinction between subject and object.
There is of course some truth in it, but I do not consider this for-
mulation to be very fortunate. For what do we mean by speaking of
an objectively existing world? This is certainly a pre-scientific
notion, never questioned by ordinary man. If he sees a dog, he sees
a dog whether it sits beside him, jumps about or runs away and
disappears in the distance as a tiny spot. All these innumerable and
vastly different sense impressions are united by an unconscious
process in his mind to the one conception dog, which remains the
same dog under all these aspects. I propose to express this by saying
that the mind constructs, by an unconscious process, invariants of
perception, and that these are what ordinary man calls real things.
And I think that science does exactly the same, only on a different
level of perception, namely using all the magnifying devices which
are the essence of observing and measuring.

The innumerable possible observations are linked again by some
permanent features, invariants, which differ from those of ordinary
perception, but are nevertheless in the same way indicators of
things, objects, particles. For in describing what we observe even
with the most refined instrument we have no other language than
the ordinary one. Thus atomistic objects have, it is true, not all the
properties of ordinary objects, but they have enough definite pro-
perties to ascribe to them physical reality of the same kind as to a
dog. I think the fact that various observations of electrons give
always the same charge, rest-mass and spin, justifies perfectly
speaking of them as real particles.

Here is another point where I disagree with EINSTEIN'S philosophy.
He accepts the doctrine of conventionalism which in my youth was
powerfully advocated by the great French mathematician HENRI
PoiNCARi. According to this view all human concepts are free
inventions of the mind and conventions between different minds,
justifiable only by their usefulness in ordinary experience. This may
be right in a restricted sense, namely for the abstract parts of
theories, but not for the connection of the theories with observations,

IO6 PHVSICS AND METAPHYSICS

with real things. It neglects the psychological fact that the building
of language is not a conscious process. And even in the abstract part
of science the use of concepts is often decided by facts, not by
conventions.

An instructive example is SCHRODINGER'S attempt to interpret
his electronic waves as a diffuse cloud of electricity, sacrificing the
particle concept. It was soon abandoned, since electrons could be
counted. The corpuscular character of the electron is certainly not
a convention.

If we thus have to attribute a definite reality to the particles,
what about the waves ? Are they also real and in what sense? It has
been said that electrons appear sometimes as waves, sometimes as
particles, perhaps changing over every Sunday and Wednesday,
as a great experimentalist mockingly remarked, obviously in a fit of
anger about the somersaults of the theorists. I cannot agree to this
view. In order to describe a physical situation, one has to use both
waves, describing a 'state', i.e. the whole experimental situation,
and particles, the proper objects of atomic research. Though the
wave functions are representing, by their square, probabilities, they
have a character of reality. That probability has some kind of reality
cannot be denied. How could, otherwise, a prediction based on
probability calculus have any application to the real world? I am
not deeply interested in the numerous attempts to make this more
understandable. It seems to me, just as the necessity of the causal
relations of classical physics, something beyond physics, a meta-
physical idea. The same holds for the wave functions of quantum
mechanics. One could call the use of particles and waves in physics
a duality in the description, which should be strictly distinguished
from complementarity.

Let us now finally ask whether these new developments in physics
have any bearing on other subjects, and principally on the great
problems of metaphysics. There is first the eternal dispute between
idealism and realism in philosophy. I do not think that the new
ways in physics can produce any weighty argument for one side or
the other. Whoever believes that the only important reality is the
realm of ideas, of the spirit, should not occupy himself with science.
The scientist must be a realist, he must accept his sense impressions
as more than hallucinations, as messages of a real outer world. In
disentangling these messages he uses ideas of a very abstract kind,
group theory in spaces of many or even infinitely many dimensions
and things like that, but finally he has his observational invariants
representing real things with which he learns to operate like any
craftsman with his wood or metaL Modern theory has made the part

PHYSICS AND METAPHYSICS IO7

of the ideas more extended and refined, but not changed the
whole situation.

But a real enrichment of our thinking is the idea of comple-
mentarity. The fact that in an exact science like physics there are
mutually exclusive and complementary situations which cannot be
described by the same concepts, but need two kinds of expressions,
must have an influence, and I think a welcome influence, on other
fields of human activity and thought. Here again NIELS BOHR has
shown the way. In biology the concept of life itself leads to a comple-
mentary alternative: the physico-chemical analysis of a living
organism is incompatible with its free functioning and leads in its
extreme application to death. In philosophy there is a similar
alternative in the central problem of free will. Any decision can be
considered on one side as a process in the conscious mind, on the
other as a product of motives, implanted in the past or present from
the outside world. If one sees in this an example of complementarity
the eternal conflict between freedom and necessity appears to be
based on an epistemological error. But I cannot enter into the
discussion of these questions which are only just beginning to be
seen in this way. Let me conclude by a remark on RUSSELL'S defini-
tion of metaphysics from which I started: that it is an attempt to
conceive the world as a whole by means of thought. Has the lesson
in epistemology which we learned from physics any bearing on this
problem? I think it has, in showing that even in restricted fields a
description of the whole of a system in one picture is impossible;
there are complementary images which do not apply simultaneously
but are nevertheless not contradictory and exhaust the whole only
together. This is, I think, a very healthy doctrine, which properly
applied may remove many violent disputes not only in philosophy
but in all ways of life. For instance, in politics. The president of the
Russian Academy of Sciences, Professor VAVILOV, has published (in
Vox) an interesting article in which he explained the ideas of
dialectical materialism and used as example the development of
optics. The thesis 'light consists of particles 5 and the antithesis light
consists of waves 5 fought with one another until they were united
in the synthesis of quantum mechanics, and the same holds for
electrons and other constituents of matter. That is very well and
indisputable. Only why not apply it to the thesis Liberalism (or
Capitalism), the antithesis Communism, and expect a synthesis,
instead of a complete and permanent victory for the antithesis?
There seems to be some inconsistency. But the idea of complement-
arity goes deeper. In fact this thesis and antithesis represent two
psychological motives and the corresponding economic forces, both

loB PHYSICS AND METAPHYSICS

justified in themselves but, in their extremes, mutually exclusive.
Complete freedom of the individual in economic behaviour is in-
compatible with the existence of an orderly state, and the totalitarian
state is incompatible with the development of the individuum. There
must exist a relation between the latitudes of freedom A/* and of
regulation Ar, of the type A/ . Arr^p, which allows a reasonable
compromise. But what is the 'political constant 9 p ? I must leave this
to a future quantum theory of human affairs. The world which is
so ready to learn the means of mass-destruction from physics, would
do better to accept the message of reconciliation contained in the
philosophy of complementarity.

PHYSICS IN THE LAST FIFTY
YEARS*

[First published in Nature, Vol. 168, p. 625, 1951.]

'T'HE following review is based on personal recollections and <
A claim historical accuracy and completeness. I shall tell yot

I cannot

I accuracy and completeness. I shall tell you what
has impressed me most, since I attended, in 1901, my first lecture at
the University of Breslau, my home city. We were taught what is
called to-day classical physics, which was at that time believed to
be a satisfactory and almost complete description of the inorganic
world. But even MAXWELL'S theory of the electromagnetic field was,
about 1900, not a part of the ordinary syllabus of a provincial
German university, and I remember well the impression of bewilder-
ment, admiration and hope which we received from the first lecture
on this subject given to us by the then young and progressive
lecturer CLEMENS SCHAEFER (still active at Cologne).

The first great event of a revolutionary character happened in
1905 with EINSTEIN'S theory of relativity. I was at that time ha
Gottingen and well acquainted with the difficulties and puzzles
encountered in the study of electromagnetic and optical phenomena
in moving bodies, which we thoroughly discussed in a seminar held
by HILBERT and MTNKOWSKI. We studied the recent papers by
LORENTZ and POINCARE, we discussed the contraction hypothesis
brought forward by LORENTZ and FITZGERALD and we knew the
transformations now known under LORENTZ'S name. MINKOWSKI
was already working on his four-dimensional representation of space
and time, published in 1907, which became later the standard
method in fundamental physics. Yet EINSTEIN'S simple consideration
by which he disclosed the epistemological root of the problem (the
impossibility of defining absolute simultaneity of distant events
because of the finite velocity of light signals) made an enormous
impression, and I think it right that the principle of relativity is
connected with his name, though LORENTZ and POINCAR should
not be forgotten.

Although relativity can rightly be regarded as the culmination of
nineteenth-century physics, it is also the mainspring of modern

* Substance of a paper read on August I3th before Section A (Mathematics
and Physics) of the British Association meeting at Edinburgh.

109

HO PHYSICS IN THE LAST FIFTY YEARS

physics because it rejected traditional metaphysical axioms, NEW-
TON'S assumption about the nature of space and time, and affirmed
the right of the man of science to construct his ideas, including
philosophical concepts, according to the empirical situation. Thus a
new era of physical science began by an act of liberation similar to
that which broke the authority of PLATO and ARISTOTLE in the
time of the Renaissance.

That result of relativity which later proved to be the most im-
portant, namely, the equivalence of mass and energy as expressed
by the formula E= me*, was at that time considered to be of great
theoretical, but scarcely of any practical, interest.

In 1913 EINSTEIN'S first attempt on general relativity became
known; it was perfected two years later. It is the first step not only
beyond Newtonian m^ta-physics, but also beyond Newtonian physics.
It is based on an elementary but so far unexplained fact that all
bodies fall with the same acceleration. To this day it is this empirical
foundation which I regard as the corner-stone of the enormous
mathematical structure erected on it. The logical way which led
from this fact to the field equations of gravitation seems to me more
convincing than even the confirmation of the astronomical predic-
tions of the theory, as the precession of the perihelion of Mercury,
the deflexion of light by the sun and the gravitational shift of
spectral lines.

EINSTEIN'S theory led to a revival of cosmology and cosmogony
on an unprecedented scale. I am not competent to judge whether
it was the theory which stimulated the astronomers to build bigger
and more powerful instruments, or whether the results obtained fc
with these, like BUBBLE'S discovery of the expanding universe,
stimulated the theoreticians to still loftier speculations about the
universe. The result, however, is undoubtedly that our astronomical
horizon to-day, in 1951, is vastly wider, our ideas about the creation
vastly grander than they were at the beginning of the period. We
can estimate the actual age of the world (some thousand millions
of years), its present size (determined by the receding nebulae
reaching the velocity of light) and the total number of nebulae, stars
and atoms, and we have good reasons for assuming that the laws of
physics are the same throughout this vast expanse. The names of
FRIEDMAN, LEMAJTRE, EDDINGTON and ROBERTSON must here be
mentioned.

But after this boast let me conclude this section on a note of
modesty. The fundamental problem of connecting gravitation with
other physical forces, to explain the strange value of the gravitational
constant, is still unsolved in spite of EDDINGTON'S obstinate.

PHYSICS IN THE LAST FIFTY YEARS III

ingenious attempts. The most promising idea seems to me that of
DIRAC, developed by JORDAN, that the gravitational constant is not
a constant at all, but a scalar field quantity, which like the other
ten, the components of the metric tensor, undergoes a secular
change and has acquired its present value in the course of time
elapsed since the creation of the universe.

Before speaking about the most characteristic features of modern
physics, atomistics and the quantum concept, I have to dwell for a
short time upon classical physics which, of course, has not suddenly
ceased to exist, but continues and flourishes to such a degree that I
should venture to say: by far the greatest part of the time and effort
of physicists is still devoted to problems of this kind, even of those,
frequently found in the United States, who believe that nuclear
research is the only decent pursuit deserving the name of physics.

In fact, the progress and success since 1900 in ordinary mechanics,
elasticity, acoustics, hydro- and aero-dynamics, thermodynamics,
electrodynamics and optics is spectacular enough. You have only to
remember that in 1900 the internal combustion engine was in its
infancy, motor-cars often brought in by horses and the aeroplane a
fantastic dream. It would be impossible to attempt even the crudest
sketch of these and other technical developments due to physics. Let
me only mention a few characteristic points.

The first is the adoption of a more realistic attitude. In the nine-
teenth century the mechanics of solids and fluids were beautiful
mathematical theories well suited for providing examination papers.
To-day, they tackle actual problems of daily life and technology,
for example, in hydrodynamics, boundary layers, heat transfer,
forces on moving rigid bodies like the wings of aeroplanes, the
stability of these, even for supersonic velocities. Among the pioneers
whom I personally knew are G. I. TAYLOR, PRANDTL, KARMAN. In
elasticity we have a similar development; the narrow field of
problems accessible to analytical solutions has been enormously
extended by numerical methods (SOUTHWELL'S relaxation method)
and the results are checked by photoelastic observations on trans-
parent models.

This trend has been strongly assisted by the invention of mechani-
cal and electrical computing machines. The speed and power of the
modern instruments based on electronic valves has stirred the
imagination of the world and given rise to a new science, cybernetics,
the advocates of which expect a revolution of human civilization
from these artificial brains a belief which I do not share.

Acoustics, the branch of elasticity dealing with the propagation of
waves, was confronted by numerous problems through the invention

U2 PHYSICS IN THE LAST FIFTY YEARS

of the gramophone, the telephone and broadcasting. Here again
the electronic valve was a powerful tool. Ultrasonic vibrations have
been used for studying the elastic properties of crystals, for signalling
and for time-keeping. The clock controlled by the oscillations of a
piezo-quartz crystal seems to be more accurate and reliable than
ordinary pendulum clocks.

Prof. ANDRADE has given an account of the origin and the develop-
ment of thermodynamics which in 1900 was considered to be com-
plete, with its two fundamental theorems (conservation of energy,
increase of entropy). But this complacent conviction was wrong
here as in many other cases.

In 1907 NERNST added a third theorem concerning the behaviour
of substances at zero temperature. Of its numerous applications to
physics and physical chemistry I can only mention the prediction of
chemical equilibria and reactions, as exemplified by HABER'S method
of fixing nitrogen from the air (1914). The experimental approach
to absolute zero made great strides. KEESOM arrived in 1931 at o*7K
with the help of liquid helium. GIAUQUE and MACDOUGALL devised
in 1933 a new method for cooling, using the demagnetization of
paramagnetic salts. The absolute scale of temperature was extended
below iK by KURTI and SIMON (1938) and others. Strange
phenomena were discovered in this region, the supraconductiviry of
metals by KAMERLINGH-ONNES in 1911, and the superfluidity of
liquid helium by KEESOM and WOLFKE in 1927, ALLEN and MEISNER,
KAPITZA and others.

Even at higher temperatures new phenomena were found, for
example, in the field of highly concentrated electrolytic solutions
where the names of BJERRUM, G. N. LEWIS, DEBYE and HUGKEL
must be mentioned.

An approach to extreme conditions from another angle was
made by BRIDGMAN (since 1905), who systematically investigated
the properties of matter under high pressure, reaching more
than 100,000 atmospheres. His latest triumph is the observation
of the breakdown of the electronic shells of alkali atoms under
pressure.

Of great importance seem to me the recent investigations started
by ONSAGER in 1930 and continued by GASIMIR, PRIGOGINE, DE
BOER and DE GROOT, by which thermodynamics is generalized so as
to apply to irreversible processes, by combining the classical laws of
flow with one single result of statistical mechanics, the so-called
principle of microscopic reversibility. The results seem to have a
bearing on the understanding of the processes going on in living

PHYSICS IN THE LAST FIFTY YEARS 1 13

The progress of electrodynamics is obvious to everybody in
technical applications : improvements in the production of power
and its transmission over long distances; telecommunication methods,
such as telegraphy, telephony and wireless transmission. In 1900
electromagnetic waves were a laboratory experiment. Since MAR-
CONI'S success in 1895 broadcasting has become a powerful factor
in human affairs.

Electromagnetic waves comprise the whole of optics, but it would
be quite impossible to give an account of the progress in all branches
of optical research and practice. The improvements and refinements
of all kinds of optical apparatus, of the experimental and theoretical
investigation of diffraction, refraction, absorption and scattering are
enormous. Let me mention only a few outstanding achievements in
spectroscopy because of their bearing on atomic physics : the dis-
covery of the ZEEMAN and STARK effect, the disentanglement of
spectral series by RYDBERG, PASCHEN, RUNGE, Rrrz and others, the
RAMAN effect, the extension of the spectrum towards the ultra-
violet and infra-red, and finally the closing of the gap, still existing
in 1900, between the longest light or heat waves and the shortest
radio waves. The pressure of war helped to develop the method
known as radar. In the laboratory it provided the magnetic reson-
ance effect, used for the study of atoms, molecules, crystals (GLEETON
and WILLIAMS, 1934; GRIFFITH, 1948), and even for the determina-
tion of nuclear spin and quadrupole moments (RABI, 1938). It has
also enriched our knowledge of the world at large by the application
to the ionosphere (APPLETON and BARNETT, BREIT and TUVE, 1925)
and to celestial bodies. Reflexions have been obtained from the
moon (U.S. Signal Corps, 1948) and from meteors (HEY and
STEWART, 1946), and waves coming from the Milky Way (JANSKI,
1931) have been observed. This new radio-astronomy will have a
profound influence on cosmology.

We now come to atomistics. Although firmly established in the
nineteenth century, there were still, in 1900, some distinguished
physicists who did not believe in atoms. To-day, such people would
be regarded as 'cranks', since the evidence for the atomistic structure
of matter is overwhelming.

There are two different but closely interwoven problems to be
answered by atomistics: (i) What is the nature of the atoms? (2)
How can the behaviour of matter in bulk be accounted for in terms
of the collective action of atoms ?

Let us begin with the latter question, as it has been answered for
a special type of matter already in the nineteenth century: I mean
the kinetic theory of gases and its extension to more general systems

114 PHYSICS IN THE LAST FIFTY YEARS

in statistical equilibrium through GIBBS' statistical mechanics. This
was in 1900 a reasonable hypothesis. But EINSTEIN'S explanation of
the Brownian movement in 1904 and SMOLUCHOWSKI'S consecutive
work in 1906 provided direct physical evidence for the correctness
of the kinetic theory and led PERRIN in 1909 to a reliable value of
the number of atoms in the gram-molecule.

The theory of compressed gases and condensation started by VAN
DER WAALS in 1873 has been much improved and modernized by
URSELL (1927), MAYER (1937) and others.

A statistical treatment of paramagnetism was given by LANGEVIN
in 1905, and extended to ferro-magnetism by WEISS in 1907. This
was the first example of a type of statistical problem dealing with
so-called order-disorder phenomena, to which, for example, the
properties of alloys belong. These methods are to-day of great
practical importance.

The logical foundations of statistical mechanics were critically
examined by PAUL and TATYANA EHRENFEST (1911) and its mathe-
matical methods vastly developed by DARWIN and FOWLER (1922).

While a satisfactory kinetic theory of liquids, in spite of great
efforts, is still lacking even to this day, our knowledge of the solid
state has been greatly increased. This work is closely connected
with research on X-rays. The nature of X-rays was controversial
until 1912. Selective absorption and polarization discovered by
BARKLA in 1909 indicated wave structure. A year later W. H. BRAGG
found evidence for corpuscular structure. In 1912 POHL and WALTER
obtained diffraction at a slit from which Sommerfeld estimated
the wave-length. The dispute was finally settled in favour of waves
when LAUE and his collaborators found, in 1912, diffraction of
X-rays by crystals, demonstrating at the same time the atomistic
nature of solids, the lattice structure of crystals, which had been
hypothetically assumed for a long time.

In the hands of W. H. and W. L. BRAGG this method opened a
new science, atomistic crystallography, which abounds in ingenious
experiments and mathematical considerations, as the systematic
application of group theory initiated by SOHNKE as early as 1879
and perfected by SCHONFLIES and FEDOROW in 1891.

Upon this empirical geometry of crystal lattices there has been
erected a dynamical theory which actually started as one of the first
applications of quantum theory with EINSTEIN'S work of 1907 on the
specific heat of solids at low temperatures, and its refinements by
DEBYE and by KARMAN and myself in 1910, which, however, has
also a large field of application in the classical domain, predicting
relations between elastic, thermal and optical properties of crystals.

PHYSICS IN THE LAST FIFTY YEARS 115

While for a time the ideal lattice was the central object of study, we
begin to-day to understand the reasons why actual crystals do in
many ways deviate from this ideal pattern.

Many of these investigations are independent of a detailed
knowledge of the atoms themselves, using only some crude averages
of their geometrical and dynamical properties, like diameter, charge,
dipole moment, polarizability, etc.

The problem that remains is to understand these averages; that
means, to investigate the nature of the atoms themselves.

The research in the structure of the atom is intimately connected
with radioactivity. The discovery of radioactivity belongs to the
nineteenth century. Its rapid development is mainly due to one
man Lord RUTHERFORD. He demonstrated the atomistic character
of the a- and ^-radiation by counting the particles, using first the
scintillation method of CROOKES (1903), later the Geiger counter
(1908). In the later development of counting methods a decisive
factor was the amplifying electronic valve, invented in its simplest
form (diode) by FLEMING in 1904 and improved (triode, pentode,
etc.) by DE FOREST in 1907 and LANGMUIR in 1915.

Let me mention here some other experimental techniques of great
importance which enable us not only to count but also actually to
see the tracks of particles: C. T. R. WILSON'S cloud chamber (1911)
and its refinement by BLAGKETT (1937), the counter-controlled
cloud chamber. Then the method of photographic tracks discovered
by BLAU and WURMBACHER in 1937, which through the improve-
ment of emulsions has become a most efficient tool for studying
atomic processes.

The first revolutionary results, obtained with the then available
primitive experimental technique by RUTHERFORD and SODDY
about 1900, were the laws of radioactive disintegration which
shattered the belief in the invariability of the chemical elements.
These laws differ from the ordinary deterministic laws of classical
physics, being intrinsically statistic and ^deterministic.

At the same time ample proof for the existence of isotopes was
found among the radioactive elements. Later, in 1913, J. J. THOMSON
discovered the first example of isotopy among ordinary elements
(neon) by electromagnetic deflexion. From here came on one hand
ASTON'S mass spectrograph (1919), the renewal of PROUT'S hypo-
thesis and the modern version of the Periodic Table with its arrange-
ment of the atoms according to nuclear charge (atomic number )
as opposed to mass (mass number A); on the other hand, the
separation of isotopes in bulk as preformed to-day on an industrial
scale for the production of fissionable material.

I j6 PHYSICS IN THE LAST FIFTY YEARS

The distinction between these two numbers and A is mainly
due to RUTHERFORD'S second great discovery (1911), the nucleus,
obtained through the observation of scattering of a-rays. The result
that COULOMB'S law is valid down to nuclear dimensions suggested
to RUTHERFORD the planetary model of the atom, with the nucleus
in place of the sun, and electrons in place of the planets. A welcome
confirmation of this was soon (1913) provided by MOSELEY with
the help of X-ray spectra. But formidable theoretical difficulties
arose because of the lack of stability of such systems according to
the laws of classical mechanics.

In fact, atomic research had reached here a point where progress
was not possible without a radical change of our fundamental
conceptions.

This revolution of thought was already in progress. It had started
in 1900, just at the beginning of this period of review, when PLANCK
convinced himself that the observed spectrum of black bodies could
not be accounted for by classical mechanics, and put forward the
strange assumption that finite quanta of energy s exist which are
proportional to the frequency v, s = Av.

The physical world received this suggestion with great scepticism
as it did not fit at all into the well-established wave theory of light.
Years passed without much happening. But in 1905 EINSTEIN took
up PLANCK'S idea and gave it a new turn; he showed that by
assuming the light to be composed of particles, later called photons,
a quantitative explanation of the photoelectric effect in metals
and* of similar phenomena is obtained. Using EINSTEIN'S inter-
pretation MILIJKAN (1910) derived from measurements of the
photo-effect a value of A in excellent agreement with PLANCK'S
original one.

Further evidence for the existence of quanta was given again by
EINSTEIN in 1907 through his theory of specific heat, mentioned
already, which not only removed some very disquieting paradoxes
of the kinetic theory, but also served as a sound foundation of the
modern theory of molecules and crystals.

The final triumph of quantum theory was BOHR'S application to
RUTHERFORD'S planetary model in 1913. It solved the riddle of
atomic stability, explained the mysterious spectral series and the
main features of the periodic system.

BOHR was, right from the beginning, quite clear that the appear-
ance of the quantum meant a new kind of natural philosophy, and
so it has turned out. Yet at the same time BOHR was anxious to
keep the connexion with classical theory as close as possible, which
he succeeded in doing with the help of his principle of correspondence.

PHYSICS IN THE LAST FIFTY YEARS I lj

There followed a period of about twelve years in which BOHR'S
ideas were confirmed and developed. Here are a few outstanding
events :

FRANCK'S and HERTZ'S experiments to demonstrate the existence
of stationary states with the help of electronic collisions (1914). The
disentanglement of the multiplet spectra, including X-rays, by
numerous authors, theoretically guided by BOHR and SOMMERFELD.
LANDERS formula for the Zeeman effect (1921) which led finally to
the suggestion of the spinning electron by UHLENBECK and GOUD-
SMIT (1925). The confirmation of SOMMERFELD'S 'quantization
of direction' by STERN and GERLACH (1921). The refinement
of the theory of the periodic system by BOHR himself, confirmed
at once by the discovery of one of the missing elements, hafnium,
by COSTER and HEVESY (1922). Then most important FAULT'S
exclusion principle (1924), which gave a theoretical foundation
to striking features of observation. Finally, the Gompton effect
(1923), which demonstrated the usefulness of EINSTEIN'S conception
of photons.

Thus the paradoxical situation had to be faced that both the
undulatory and the corpuscular theory of light were right in fact,
PLANCK'S formula s = Av states a relation between these contra-
dictory hypotheses.

This challenge to reason came to a climax through DE BROGLIE'S
famous thesis of 1924 in which this duality wave-corpuscle was, by
a purely theoretical argument, extended to electrons. The _ first
confirmation was given by ELSASSER (1927) with the help of experi-
ments on electron scattering on metals made by DAVISSON and
GERMER (1927), and soon these authors, and independently G. P.
THOMSON, the son of the discoverer of the electron as a particle,
produced diffraction patterns with metal foils which established
definitely the existence of DE BROGLIE'S waves.

May I mention here in parenthesis that the idea of the electron
microscope is considerably older than this theory; it was first
suggested in 1922 by H. BUSCH on the grounds of considerations
analogous to geometrical optics. After DE BROGLIE the wave theory
of optical instruments became applicable and the resolving power
could be determined. I cannot dwell on details, but I wish to remind
you that to-day not only bacteria and viruses but even big molecules
can be made visible and photographed.

The duality wave-corpuscle made an end of the naive intuitive
method in physics which consists in transferring concepts familiar
from everyday life to the submicroscopic domain, and forced us to
use more abstract methods.

I j# PHYSICS IN THE LAST FIFTY YEARS

The first form of this new method was mainly based on spectro-
scopic evidence which led KRAMERS and HEISENBERG to the convic-
tion that the proper description of the transition between two
stationary states cannot be given in terms of the harmonic com-
ponents of these states separately, but needs a new kind of transition
quantity, depending on both states. HEISENBERG'S quantum
mechanics of 1926 is the first formulation of rules to handle these
transition quantities, and these rules were soon recognized by myself
as being identical with the matrix calculus of the mathematicians.
This theory was developed by HEISENBERG, JORDAN and myself,
and independently, in a most general and perfect manner, by DIRAG.

Again, independently SGHRODINGER developed in 1926 DE
BROGLIE'S wave mechanics by establishing a wave equation valid not
only for free electrons but also for the case of external fields and
mutual interaction, and showed its complete equivalence with
matrix mechanics.

Concerning the physical interpretation, SGHRODINGER thought
one ought to abandon the particle conception of the electron
completely and to replace it by the assumption of a vibrating con-
tinuous cloud. When I suggested that the square of the wave function
should be interpreted as probability density of particles, and pro-
duced evidence for it by a wave theory of collisions and other
arguments, I found not only SCHRODINGER in opposition, but also,
strangely enough, HEISENBERG. On the other hand, DIRAG developed
the same idea in a mathematically brilliant way, which was soon
generally accepted, also by HEISENBERG who produced a most
important contribution by formulating his uncertainty relations
(1927). These paved the way for a deeper philosophical analysis of
the foundations of the new theory, achieved by BOHR'S principle of
complementarity, which replaces to some degree the classical concept
of causality.

In a very short time the new theory was well extablished by its
successes. I can mention only a few points : PAULI'S matrix representa-
tion of the spin and DIRAC'S relativistic generalization (1928) which
led to the prediction of the positron, actually found by ANDERSON
in 1932. Then came the systematic theory of the electronic structures
of atoms and molecules and their relations to line and band spectra,
to magnetism and other phenomena. WIGNER showed in 1927 how
the general features of atomic structures could be found with the
help of group theory. HARTREE, FOGK, HYLLERAAS and others
developed numerical methods. The theory of collisions of atoms
with electrons and other atoms was started by myself and developed
by BETHE, MOTT, MASSEY and others, from which finally sprang a

PHYSICS IN THE LAST FIFTY YEARS I IQ

general theory of the penetration of particles through matter by BOHR.

Further, the derivation of the nature of the chemical bond,
initiated by HEITLER and LONDON in 1927, was worked out by
HUND, SLATER, MULLIKEN, PAULING and others. Even the compli-
cated phenomena of reaction velocities, including catalytic accelera-
tion, have been reduced to quantum mechanics.

Finally came DIRAC'S most important theory of emission, absorp-
tion and scattering of electromagnetic radiation which led to the
first systematic attempt of formulating quantum electrodynamics by
FERMI, JORDAN, HEISENBERG and PAUU (1929), and later to the
general theory of quantized fields and their interaction ( WENTZEL,

ROSENFELD, from 1931).

The last period of our fifty years is dominated by nuclear physics.
Although the importance of nuclear research is probably greater
than that of any other branch of physics, I shall be rather short
about it for it is the most recent phase of our science and scarcely
yet history.

The first breaking up of a nucleus was achieved by RUTHERFORD
in 1919, by bombarding nitrogen with a-rays. Artificially accelerated
particles were first used by COCKCROFT and WALTON in 1930. At
that time the nucleus was believed to be composed of protons and
electrons. But this led to difficulties if one tried to derive the angular
momenta of nuclei from the spins of the component particles. In 1932
GHADWICK discovered the neutron, and those difficulties disappeared
if the nucleus was assumed to consist of protons and neutrons, or
charged and uncharged 'nucleons'. FERMI showed in 1932 that
neutrons are most efficient in disrupting nuclei as they are not
repelled by the nuclear charge. Many of the residual nuclei were
found by IRENE and FREDERIC JOLIOT-GURIE in 1934 to be radio-
active themselves.

The continuous yff-ray spectrum offered great difficulties to the
understanding until PAUU, in 1931, suggested the existence of the
neutrino and FERMI developed, in 1934, the neutrino theory of the
/?-decay where the laws of conservation of energy and momentum
are preserved. The line spectrum of /?-rays was recognized to be of
secondary origin, namely, due to the expulsion of electrons from the
electronic cloud by y-rays emitted by the nucleus.

The need for fast projectiles was first supplied by the use of cosmic
rays. These had been discovered by HESS already in 1912 and their
study has grown to-day into a vast science, covering not only
nuclear physics but also geophysics, astronomy and cosmology.

The artificial production of fast particles has made enormous strides
through the construction of powerful accelerating machines, as that

120 PHYSICS IN THE LAST FIFTY YEAfeS

of VAN DE GRAAFF (1931), LAWRENCE'S cyclotron (1931), KERST'S
betatron (1940) and combinations of these, like the synchrotron.

The clue to the interpretation of nuclear transformations is
EINSTEIN'S formula E = me*, or more precisely, the relativistic
conservation laws of energy and momentum. I am not an expert in
the new awe-inspiring science of nuclear chemistry and shall make
no attempt to describe it. I can say only a few words on the theoreti-
cal problems of nuclear physics. It is remarkable how many
important facts can be understood by extremely simple models, as,
for example, GAMOW'S crater model (1928)3 which explains the
a-decay and the GEIGER-NUTTAL relation between cc-energy and life-
time; and the liquid-drop model, suggested by VON WEIZSAGKER in
r 9 2 5> to explain the mass defect (nuclear energy) curve and later
used successfully by BOHR (1935) to explain the mechanism of
capture, re-emission and fission. A great amount of work has been
done on exact quantum-mechanical calculations of the structure and
properties of light nuclei (in particular the deuteron) and of the
effect of collisions, with the aim of learning something about the
nuclear forces. Important results have been, obtained, but altogether
the situation is not satisfactory.

Quite independently of detailed theories, the empirical values of
the nuclear masses (internal energies) indicate that the light nuclei
have the tendency to fuse, the heavy ones to disintegrate; hence all
matter, except the elements in the middle of the periodic system
(iron), is in principle unstable. But reaction velocities are, under
terrestrial conditions, so extremely slow that nothing happens. It is,
however, different in the interior of stars; BETHE showed in 1938
that one can account for the heat developed in the sun and the stars
by a nuclear catalytic chain reaction, the fusion of four nucleons to
form a helium nucleus.

The opposite phenomenon, the fission of the heavy nucleus of
uranium into almost equal parts, discovered by HAHN and STRASS-
MANN in 1938, has initiated a new era in the sociological situation of
our science and very likely in the history of mankind. Here is a list
of events:

The establishment in 1939 of the possibility of a self-supporting
chain reaction by different authors (JouoT, HALBAN and KOVARSKI;
FERMI; SZILLARD) ; the construction of the first nuclear reactor or
'pile' under the direction of FERMI in 194.2, and, finally, the harness-
ing of the industrial power of the United States to produce the
atom bomb.

The political and economic implications of this development are
too formidable to be discussed here; but I cannot refrain from saying

PHYSICS IN THE LAST FIFTY YEARS 121

that I, personally, am glad not to have been involved in the pursuit
of research which has already been used for the most terrible mass
destruction in history and threatens humanity with even worse
disaster. I think that the applications of nuclear physics to peaceful
ends are a poor compensation for these perils.

However, the human mind is adaptable to almost any situation,
So let us forget for a while the real issues and enjoy the useful results
obtained from the pile. In physics the remaining few gaps of the
Periodic Table have been filled and five or six transuranium elements
(among them fissionable nuclei like neptunium and plutonium)
discovered. Innumerable new isotopes of known elements have been
produced. Some of these can be used as 'tracers' in chemical and
biological research as first suggested by VON HEVESY in 1913; others
as substitute for the expensive radium in industrial research and in
the treatment of cancer.

From the point of view of natural philosophy the most important
achievement of the past decade seems to me the discovery of the
meson, theoretically predicted by YUKAWA in 1935, which showed
how far we are still removed from a knowledge of the real funda-
mental laws of physics. YUKAWA became convinced that the forces
between nucleons are at least as important as the electromagnetic
forces, and by applying the field concepts in analogy to MAXWELL'S
theory was able to predict a new particle which has the same rela-
tion to the nuclear field as the photon to the electromagnetic field,
but has a finite rest-mass, which from the range of nuclear forces
could be estimated to be about 300 electron masses. Soon the
existence of mesons was experimentally confirmed in cosmic rays
by ANDERSON and NEDDERMEYER in 1936 and later with particles
produced by the cyclotron in California in 1948. The method of
photographic tracks has, in the hands of POWELL (from 1940) and
others, produced a wealth of new results, for example, the sponta-
neous disintegration of the meson of about 300 electron masses
into a lighter one of about 200 electron masses and a neutral particle.
A meson of about 900 electron masses has been fairly well established,
and it is not unlikely that still more types exist.

It is obvious that to understand all this a much deeper research
in the theory of quantized fields and their interaction is necessary.
A revised and modernized quantum electrodynamics was published
independently by SCHWINGER in the United States and TOMONAGA
in Japan in 1947, and from this has sprung a considerable amount
of literature, aiming at the elimination of divergence difficulties and
calculating effects of higher order, inaccessible to the older theory.
A great success was the, explanation of an observation made by

!22 PHYSICS IN THE LAST FIFTY YEARS

LAMB and RETHERFORD in 1947, which showed that DIRAC'S
celebrated theory of the hydrogen spectrum is not quite correct. But
it becomes more and more clear that all these mathematical refine-
ments do not suffice, and that a far more general theory has to be
found s in which a new constant (an absolute length or time, or
mass) appears and which ought to account for the masses found in
Nature. I wish to end this outlook into the future with a remark
I have recently heard from HEISENBERG. We have accustomed
ourselves to abandon deterministic causality for atomic events ; but we
have still retained the belief that probability spreads in space (multi-
dimensional) and time according to deterministic laws in the form
of differential equations. Even this has to be given up in the high-
energy region. For it is obvious that the absolute time interval
restricts the possibility of distinguishing the time order of events. If
this interval is defined in the rest system it becomes large in a fast-
moving system according to the relativistic time expansion (in
contrast to the contraction of length). Hence the indeterminacy of
time order and therefore of cause-effect relation becomes large for
fast particles.

Thus experience again leads us to an alteration of the meta-
physical foundations of a rather unexpected kind. In fact, traditional
philosophy has provided the leaders of our science, like EINSTEIN,
BOHR and HEISENBERG, with problems in so far as it failed to supply
answers agreeing with experience. I am convinced that although
physics free from metaphysical hypotheses is impossible, these
assumptions have to be distilled out of physics itself and continuously
adapted to the actual empirical situation. On the other hand, the
continuity of our science has not been affected by all these turbulent
happenings, as the older theories have always been included as
limiting cases in the new ones. The scientific attitude and the methods
of experimental and theoretical research have been the same all
through the centuries since GALILEO and will remain so.

THE CONCEPTUAL SITUATION IN PHYSICS

AND THE PROSPECTS OF ITS FUTURE

DEVELOPMENT

[37th Guthrie Lecture, delivered March I3th, 1953. First published in Proc.
Pkys. Soc., A, Vol. LXVI, pp. 501-513, 1953.]

T ET me begin with a personal remark. Fifty years ago I was a
-*- J young student of science, in my second academic year. At that
time PLANCK'S radiation formula and the quantum hypothesis
were already more than two years old. But I was ignorant of those
momentous events. We were taught NEWTON'S mechanics and its
applications, and we were cautiously introduced to MAXWELL'S
theory of the electromagnetic field.

To-day the situation may be similar. A great discovery may be
made somewhere by somebody of which I have heard nothing or
whose importance I do not see. With increasing age it becomes more
and more difficult to keep step with contemporary research. My
knowledge of what is going on in the laboratories and studios all
over the world is now almost as scanty as it was half a century ago.
Yet the years have not passed without trace. They have left an
accumulation of experience over a wide horizon, and this encourages
me to speak to you about my impressions of the present situation in
theoretical physics and the direction in which it is moving. A fore-
cast about the future may appear presumptuous, for science has
always been full of surprises, unexpected experimental results
which changed the structure of the theory. Yet I venture certain
guesses because of a phenomenon which might be called the 'stability
of the principles 9 . 1 do not suggest that, apart from mathematics, there
are any principles which are unchangeable, a priori in the strictest
sense. But I think that there are general attitudes of the mind which
change very slowly and constitute definite philosophical periods
with characteristic ideas in all branches of human activities, science
included. PAULI, in a recent letter to me, has used the expression
'styles', styles of thinking, styles not only in art, but also in science.
Adopting this term, I maintain that physical theory has its styles
and that its principles derive from this fact a kind of stability. They
are, so to speak, relatively a priori with respect to that period. If you
are aware of the style of your own time you can make some cautious

123

124 THE CONCEPTUAL SITUATION IN PHYSICS

predictions. You can at least reject ideas which are foreign to the
style of your time.

I shall not attempt a historical review of physics from this stand-
point, nor an investigation of the question whether the style of
science, in particular of physics, depends on other conditions, for
instance economic ones. I shall just begin with the modern era,
with GALILEO and NEWTON, and stress solely one characteristic
point, namely, the separation of subject and object in the descrip-
tion of natural phenomena. For the Greek philosophers the cause of
motion, the force producing the motion, was inseparable from a
living being, man or god, who felt the exertion. Moreover, they used
ideas of value as a principle of explanation. The planets moved in
circular (or epicyclic) orbits because the circle is the most perfect
curve. Perfection reigned in the celestial spheres, corruption on the
terrestrial level; law and order among the stars, chaos and strife on
earth. The Christian era introduced new ideas and is certainly a
separate period with its own style, but in regard to science it relied
on the ancients and preserved the anthropocentric, subjective
attitude. The idea of perfection was now personified in God. Natural
phenomena happen to glorify Him, to punish the wicked, to reward
the good. This motive is still strong in KEPLER.

The break came with GALILEO and NEWTON. They introduced
the disinterested, objective description and explanation which is
characteristic for the modern epoch. But the ancient style did not
disappear at once. Traces survived a long time, for instance in the
metaphysical interpretation of the minimum principles of mechanics.
MAUPERTUIS certainly believed that the minimum of action was the
expression of a purpose of Nature or the Creator. Even EULER'S
writing, where the first rigorous formulation of the principle of least
action is given, is not free from this metaphysical attitude. It finally
disappears in LAGRANGE'S work.

From now on the world is a mechanism, ruled by strict determin-
istic laws. Given the initial state, all further development can be
predicted from the differential equations of mechanics. The mini-
mum principles are not due to nature's parsimony but to human
economy of thinking, as MAGH said; the integral of action condenses
a set of differential equations into one simple expression.

The supposition is that the external world, the object of natural
science, and we, the observing, measuring, calculating subjects, are
perfectly separated, that there is a way of obtaining information
without interfering with the phenomena.

This is the philosophy of science in which we, of the older genera-
tion, have grown up. It can be called the Newtonian style, as it is

THE CONCEPTUAL SITUATION IN PHYSICS 125

modelled on NEWTON'S celestial mechanics. It was extremely success-
ful also in terrestrial matters, even when it was extended from
mechanics of material systems to electrodynamic phenomena in vacuo
and in matter. MAXWELL'S theory takes the polarity between subject
and object for granted and is strictly deterministic.

A new era, a new style, commenced in 1900, when PLANCK pub-
lished his radiation formula and the idea of the quantum of energy.
Its way was prepared by a long development which revealed the
inadequacy of classical mechanics to deal with the behaviour of
matter. The differential equations of mechanics do not determine a
definite motion, but need the fixation of initial conditions. For
instance, they explain the elliptic orbits of the planets, but not why
just the actual orbits exist. But there are regularities concerning
the latter: BODE'S well known rule. This is regarded as a question of
the prehistory of the system, a problem of cosmogony, and still
highly controversial. In the realm of atomistics the incompleteness
of the differential equations is even more important. The kinetic
theory of gases was the first example to show that new assumptions
had to be made about the distribution of the atoms at a fixed instant,
and these assumptions turned out to be more important than
the equations of motions; the actual orbits of the particles do not
matter at all, only the total energy which determines the observable
averages. Mechanical motions are reversible, therefore the explana-
tion of the irreversibility of physical and chemical processes needed
new assumptions of a statistical character. Statistical mechanics
paved the way for the new quantum era.

With the quantum came a new attitude to the polarity subject-
object. It is neither essentially subjective, as the ancient and medi-
aeval doctrines, nor wholly objective, as the post-Newtonian
philosophy.

The change was due to the breakdown of all attempts to under-
stand atomic phenomena from the standpoint of ordinary mechanics.
A new atomic mechanics had to be found, and the way leading to
it proceeded in steps. The most important of these was BOHR'S idea
of stationary states and transitions between them. The states are
certain mechanical orbits picked out by simple quantum rules, and
the energies lost or gained by the transition are connected with
frequencies of emission and absorption by PLANCK'S quantum law
E hv. The amazing success of this theory in explaining the
stability of atoms, the structure of atomic and molecular spectra,
the periodic system of elements and of many other properties of
matter did not delude BOHR into believing that this was a final
solution. He stressed, from the very beginning, the new features of

126 THE CONCEPTUAL SITUATION IN PUYSICS

the scheme, namely the indeterministic character of the transitions,
the appearance of chance in the elementary processes. This means
the end of the sharp separation of the object observed and the
subject observing. For chance can be understood only in regard to
expectations of a subject.

After twenty-five years of struggle a satisfactory theory was
obtained, from different sources. One approach, which expresses
BOHR'S ideas in a logically consistent way, is due to HEISENBERG,
the so-called matrix mechanics. Another quite independent approach
was found by DE BROGUE and developed in SCHRODINGER'S wave
mechanics. In the form given to the theory by DIRAC it is a structure
of great beauty and perfection, but rather abstract. It has been
supplemented by a doctrine of measurement, due to HEISENBERG
and BOHR, which connects the formalism with the experimental
reality.

The essential feature is that the physical quantities or, in DIRAG'S
terminology, 'observables', like coordinate, momentum, energy of a
particle, components of field strength, etc., are not represented by
variables, but by symbols with a non-commuting multiplication
law, or, more concretely, by operators A, which operate on a
quantity ft, transforming it into another quantity Aft. This function
ft is a generalization of DE BROGUE'S and SCHRODINGER'S wave
amplitude and defines the state of the system. It satisfies an equation
of the deterministic type current in classical theory. Nevertheless it
does not allow deterministic predictions about the observables, but
only statistical ones: | ft | 2 is the probability of the state repre-
sented by ft, and the expectation value of an observable A in this
state can be expressed in terms of ft. In particular, the accuracy Sq
of a measurement of a coordinate q (properly defined through the
expectation value of the mean square deviation) and the accuracy
Sp for the corresponding momentum p are found to satisfy HEISEN-
BERG'S uncertainty relation SqSp>h, where h is PLANCK'S constant.
Similar relations hold for other pairs of 'conjugate' variables.

In this abstract formulation the words particle, co-ordinate,
momentum, etc., are used, but obviously with a different meaning
from ordinary language. A dust particle is supposed to have at a
given instant a certain position and velocity. An electron or other
particle obeying the laws of quantum mechanics behaves differently
for according to the uncertainty rule a definite position (Sq very
small) demands a large Sp (>hl$q), hence a large uncertainty of
velocity. This question has been discussed so often that I need not
dwell upon it. The further development of quantum mechanics has
revealed more features of strange behaviour, for instance the lack

TIIE CONCEPTUAL SITUATION IN PHYSICS 127

of individuality of particles, which has very direct and decisive
consequences for statistical thermodynamics.

Therefore the question arises how these new conceptions of
particles and their properties can be handled without coming into
conflict with the obvious fact that the instruments used in experi-
menting with them and observing them are ordinary bodies which
obey Newtonian laws. This is the object of BOHR'S theory of measure-
ment. The essence of quantum mechanics, stripped of all mathe-
matical refinement, is the laws of PLANCK and of EINSTEIN-DE
BROGUE, namely E = hv> p = A/c; here E, p are the energy and
momentum of a particle, v, K the frequency and wave number of
the 'corresponding' wave. If one tries to visualize the meaning ^of
this correspondence in space and time, one finds a paradoxial
situation. For E, p refer to an extensionless particle, v, K to a har-
monic wave which by its very definition is infinitely extended in
time and space. The solution of the paradox must therefore be
found in an analysis of the use of the concepts of location and
duration in connection with a train of waves.

One is accustomed to apply the idea of a definite time interval
or duration to any ordinary pair of events (e.g. the fall of a stone
from my hand to the earth). Yet there are seemingly harmless cases
where this is not justified. The sentence c a musical tone lasts a
definite time 5 has no rigorous meaning. That is not a purely logical
statement, but one of fact. Indeed, a sharp staccato on the low pipes
of an organ sounds badly. For a wave train starting harmonically
but broken off at a time not large compared with the period of
vibration is not actually harmonic but a superposition of harmonic
waves of different frequencies, a wave packet: acoustically ^ a noise.
This fact is also well known in optics, where it is the basis of the
theory of the resolving power of instruments, and it has recently
become most important in the theory of information (obtained by
transmitting electromagnetic or other waves).

Elementary considerations on the mutual limitation of St and
8v, Sx and &c, lead to the relations St 8v > i, $x SK > i. They are
the root of the uncertainty rules of HEISENBERG; for if they are
multiplied by h and the PLANCKHDE BROGUE relations used, the
result is 8t SE> h, dx Sp > h. This consideration in no way mitigates
the paradoxical, almost irrational, character of the PLANGK-DE
BROGUE correspondence. But it helps to handle it in such a way
that contradictions between the results of measurement cannot

occur.

Location and duration can be measured only with the help of
rigid scales and clocks; energy and momentum only with the help

128 THE CONCEPTUAL SITUATION IN PHYSICS

of mobile parts, which react according to the conservation laws.
Thus the reciprocal uncertainty can be traced to two types of
mutually exclusive but complementary experiments. BOHR has
illustrated this 'complementarity 5 by many instructive examples,
some of these in response to attacks made by EINSTEIN, who hoped
to disprove the uncertainty rules by ingenious experimental arrange-
ments. I think that attempts against the uncertainty laws will cease
in time. The lasting result of BOHR'S endeavours is the simple
consideration given above, which shows with irrefutable logic that
the PLANCK-DE BROGUE laws of necessity imply the duality particles-
waves and the complementary quality of experimental arrangements
set up to measure 'conjugate 9 pairs of quantities, like energy-time,
momentum-location.

Intimately connected with this duality is the polarity subjective-
objective. For if an experiment must be set up in a definite way to
investigate one or the other of a conjugate pair of quantities, it is
impossible to obtain information of the system considered as such;
the observer has to decide beforehand which kind of answer he
wants to obtain. Thus subjective decisions are inseparably mixed
with objective observations. The same can be seen from the mathe-
matical description with the help of the state-function ^, which is
only determined by the whole system, including the means of
observation which depend on the subject.

This is a sketch of the modern style of physics which is accepted
by practically the whole community of experimental and theoretical
physicists. It fits exactly into the practice of electronics, spectroscopy,
radioactivity, nuclear physics and also chemistry and astrophysics.
The questions for which the theory offers answers are just those
which the experimentalist wants to be answered. He is entirely
indifferent to orbits of electrons in atoms, of atoms in gases, of
nucleons in nuclei; he is quite content with stationary states and
collision cross-sections which the theory supplies.

I think that this mode of scientific thought is also in conformity
with the general trend of contemporary philosophy. We have lost
confidence in the possibility of separating knowledge from decision,
we are aware of being at every moment spectators and actors in the
drama of life. BOHR himself has indicated generalizations of his
'complementary 5 idea to biology and psychology; ancient problems
like that of the relation of matter and mind, freedom and necessity,
are thus seen from a new angle. I cannot enter into these deep
questions, but may mention some fascinating books by VON WEIZ-
SACKER (1949, 1951), where they are treated with competence and
good taste.

THE CONCEPTUAL SITUATION IN PHYSICS 129

I venture the prediction that this style of thinking will last, and
that a future change, when it comes, will not lead back to the past,
so-called classical, style but to something more removed from it. My
confidence in this forecast rests not only on the success of the present
theory but in my personal affinity for its philosophy.

However, this view is strongly contested, just by some of those
who have done most to develop quantum theory. PLANCK himself
was sceptical. For instance, when he, as President of the Berlin
Academy, inaugurated SCHRODINGER (who was his successor to his
chair), he praised him as the man who had re-established determin-
ism through his wave equation. EINSTEIN, who renewed the cor-
puscular idea in optics, who introduced the transition probabilities
between two stationary states and is guilty of other anti-classical
deviations, has turned with a kind of passion against the statistical
interpretation of quantum mechanics. I have already mentioned
his attempts to disprove the uncertainty laws by ingenious contrap-
tions and BOHR'S refutations of these attacks. When EINSTEIN could
not maintain the existence of logical flaws in quantum mechanics he
declared it to be an 'incomplete' description of nature. I have used
the same expression before in regard to the differential equations of
classical mechanics which are incomplete without initial values for
which classical theory gives no law and which, in my opinion, lead
to absurd consequences. Imagine JV particles fixed in random
positions and another particle fired amongst them, colliding and
recoiling, according to classical laws. It is obvious that for large N
the tiniest deviations of the initial motion produce not small changes
in the final position, but an enormous variety of large effects. If all
particles are moving like gas atoms this would hold a fortiori. Thus
the supposed determinism is an illusion.

This group of distinguished men, to whom VON LAUE may be
added, may be called philosophical objectors, or, to use a less
respectful expression, general grumblers.

There are those who, aware of the unavoidable consequences of
the PLANCK-DE BROGUE relations E hv^p hx, want to sacrifice
these and preserve only one side of the picture. There are the
particle defenders or />-totallers, and the ^-wave defenders or ft-
totallers. They are of course all theoretical physicists, and you find
them well represented in a recently published book* dedicated to
DE BROGLIE on the occasion of his Goth birthday (1952). DE BROGUE
himself, though the discoverer of the electron waves, has made

* This book contains the literature in more complete form than is given here.
In the list of references at the end of this article papers are mentioned which are
quoted in the text only by e and others'.

130 THE CONCEPTUAL SITUATION IN PHYSICS

serious attempts to save determinism by introducing concealed
parameters. One of his suggestions (DE BROGLIE 1926, 1927) was to
write a complex ^-function in the form j/r = R exp (zXD) ; then SCHRO-
DINGER'S wave equation is equivalent to a set of classical equations
of motion of particles under the action of two forces, one with the
potential O, the other with a supplementary potential U. The latter
depends on R and is subject to strong fluctuations due to the inter-
action of the particles, thus producing the same effect as the un-
certainty in the current interpretation. A similar suggestion has
been independently made by MADELUNG (1927). Recently such
considerations have been renewed and refined by FRENKEL (1950,
1951) and BLOKHINTZEV (1950, 1951) in Russia, and by BOHM
(1952) in America. Already in 1932 VON NEUMANN had shown that
it is impossible to introduce concealed parameters without conflict
with confirmed results of the current theory. Therefore BOHM is
anxious to show that in the frame of present knowledge his concealed
parameters cannot be determined by experiment; he hopes that
future discoveries will make this possible. But PAULI, in the DE
BROGLIE volume mentioned, has shown that this attitude leads to
contradictions; for in problems of statistical thermodynamics the
concealed parameters must necessarily show their existence and
produce secular distortions of the BOSE- or the FERMI-DIRAC
distribution.

Thus the reactionary ^-totaller movement can be discarded.

SCHRODINGER has, right from the beginning, taken the opposite
standpoint: the whole of physics is wave theory, there are no
particles, no stationary states and no transitions, only waves. I have
already mentioned that PLANCK welcomed this idea; but the majority
of physicists continued to use the particle image and to speak of
atoms, electrons, nuclei, mesons etc.

Recently SCHRODINGER (1952) has taken up his purification
campaign and pleaded passionately for ejecting not only particles,
but also stationary states, transitions, etc., from physics. The motive
for his discovery of wave mechanics was his violent dislike of BOHR'S
instantaneous 'quantum jumps', and we can understand his triumph
when he could represent all these 'absurdities 3 in terms of well-known
and innocuous resonance phenomena of waves.

I myself might have a similar motive to declare matrices as the
only real thing. Allow me to indulge in a personal reminiscence.
When HEISENBERG published the fundamental paper in which he
cleared quantum theory from classical remnants and formulated it
in terms of transition amplitudes, he was my assistant, very brilliant
but very young, not very learned. In fact he did not exactly know

THE CONCEPTUAL SITUATION IN PHYSICS 131

what a matrix was, and as he felt stuck he asked for my help. After
some effort I found the connection with the matrix calculus, and
I remember my surprise when HEISENBERG'S quantum condition
turned out to be the matrix equation qp pq = ik. If HEISENBERG
were here instead of myself he would tell you the same story. The
matrix form of quantum mechanics was first published by myself
in collaboration with my pupil JORDAN.*

However, I have not, and never had, a particular preference for
the matrix method. When SGHRODINGER'S wave mechanics appeared
I felt at once that it demanded a non-deterministic interpretation,
and I guessed that | i/r [ 2 was the probability density; but it took
some time before I had found physical arguments in favour of this
suggestion, namely collision phenomena and transitions under
external forces. Now the strange thing happened that HEISENBERG
first disagreed and accused me of treason against the spirit of mairix
mechanics. But he soon came round and produced the wonderful
reconciliation of particles and waves with the help of his uncertainty
relation.

But now I have to return to SGHRODINGER'S attack against
particles and quantum jumps. It cannot be proved wrong, for the
^r-function which can be represented as a wave in a multi-dimen-
sional space contains all physical information provided you know
how to connect it with experience. And there is the difficulty. We
have no other language to describe what we do and what we see in
experimenting than in terms of bodies and their movements.
SCHRODINGER himself cannot avoid the particle language even when
he tries to demonstrate the supremacy of the wave language. I have
dealt with this question in detail at another place (1953) and need
not repeat it. I think SCHRODINGER'S suggestion is impracticable and
against the spirit of the time.

Yet I do not wish to create the impression that I believe the
present interpretation of quantum theory to be final. I only think
that a return to Newtonian determinism is impossible.

I have now arrived at the point where I have to make good my
promise to try some forecast of the future.

The fundamental problems of contemporary physics are con-
cerned with elementary particles and the corresponding fields, in
particular the explanation of stability or instability, masses, spin
character, interactions, etc. This is a wide programme which
includes the whole of nuclear physics and the study of cosmic rays,

* The early phase of quantum mechanics is misquoted nearly everywhere in the
literature. I have given a few more examples in my book Natural Philosophy of
Cause and Chance (Oxford: Clarendon Press, 1949), App. 27, p. 188.

132 THE CONCEPTUAL SITUATION IN PHYSICS

and it leads definitely beyond the scope of current quantum
mechanics, for the problem of the elementary masses is connected
with the difficulty of the self-energy of particles. It is well known
that the self-energy of an electron is infinite even in the classical
theory of MAXWELL-LORENTZ. In quantum theory this primary
infinity of the type e z ja (e charge, a radius, limit 0->o) is superposed
by a variety of other divergent integrals. I have followed these
investigations only from afar, but my impression is that through the
work initiated by TOMONAGA (1946) and SCHWINGER (1948) a kind
of solution has been found: By a profound mathematical method
called *renormalization ? the actual, intrinsic singularities can be
separated out and, if infinite, omitted in a way which is uniquely
fixed by postulating relativistic invariance, and the remaining
formulae give definite, finite results. DIRAG (1951) wrote about this
theory: c lt is an ugly and incomplete one, and cannot be considered
as a satisfying solution of the problem of the electron', and he
suggests an alternative theory. I think the first part of his judgment
too hard, for it is a great achievement to have a working formalism
which in the hands of the initiated leads to the explanation of such
delicate effects as the LAMB-RETHERFORD shift (1947) in the hydro-
gen terms and deviations from LANDE'S magnetic factors, etc. But I
subscribe to the view that this theory is incomplete and circumvents,
instead of attacking, the actual problem. DIRAG has suggested an
alternative theory whose main idea is that the occurrence of charge
in finite quanta, electrons, must be a quantum effect; hence the
corresponding classical theory should be a pure wave theory. By a
slight modification of the current formulae he obtains such a wave
theory, but so far he has not -succeeded in quantizing it. It is possible
that a satisfying theory of the electromagnetic field and its charges
can never be obtained because photons and electrons cannot be
treated without regard to other particles.

The most conspicuous feature of modern physics is the discovery
of more and more unstable particles, called mesons. For practical
purposes linear wave equations for each type of particle are estab-
lished with non-linear coupling terms between them. It is clear that
this is a preliminary approach which one day will have to be
superseded by a coherent theory of matter, in which the different
masses of the particles appear as eigenvalues of operators or solutions
of equations. It is now generally accepted that this theory will
contain an absolute length 0, or an absolute momentum b = hja,
and that in domains of the dimension a geometry may become
meaningless. A remarkable attempt to formulate such a situation is
due to YUKAWA (1949); he regards a field component <j> not as a

THE CONCEPTUAL SITUATION IN PHYSICS 133

function of the space co-ordinates and time, x, jy, , t, but both <j>
and #, y, z, t as non-commuting quantities, and postulates certain
commutation laws between them which are generalizations of
the current differential equations and go over into them if all
distances are large compared with the absolute length a. YUKAWA,
M0LLER (1951), RAYSKI (1951) and others have shown that the
divergences of the self-energy and other such difficulties can thus be
avoided.

The first who clearly saw the necessity of uniting the theories of
different particles was EDDINGTON. But at this time there were only
two kinds known, protons and electrons. Thus the discoveries of
mesons have made his attempt rather obsolete, quite apart from
the rather fantastic foundations. [His main assumptions led to the
integral value 137 of the reciprocal fine structure constant i/a
= hcje 2 , which is almost but not quite in agreement with the latest
observations, from which the value i/a = 1 37*0364 0*0009 is
derived (Du MOND and COHEN 1951).]

I cannot deal with the many attempts to unify the different fields.
Most of them can be reduced to the following scheme:

The wave equation (Q + m 2 )^ 1 = o (D is the d'Alembertian
operator) is replaced by/(C)^ = o, where/(g) = (S i)( 2)
. . . . ( g n ) is a polynomial of degree n; it describes the motion
of 72 independent particles with masses m x = y^ . . . . m n Vw
By using instead of D DIRAG'S operator one can take account of the
spin.. Theories of this type have been derived by BHABHA (1945)
and others from considerations of particles with higher spin. I have
suggested another way to determine the function/() which connects
this problem with that of the infinities. One can add to /() a
transcendental factor without zeros. If one has, for instance, the
differential operator in the domain of one variable q, p 2 m 2 ,
where p = ikSjdq, one can add the factor exp ( |^ 2 ) (where
b = h\a is taken as unit for p}. This has, in the first place, the
consequence that the possible momenta are cut off, thus removing
infinities. And secondly, one can determine the mass m by giving
the expression (p* m*) exp( J/> 2 ) a proper meaning; it is the
second Hermitian function of p for m 2 = 2, hence identical with
its FOURIER transform. This remark suggests the application of the
general principle that the whole of physics can be formulated in
terms of transformation groups and their invariants. By postulating
reciprocal invariance (i.e. against FOURIER transformation) it seems
to be possible to determine a set of masses as the roots of (Hermitian)
polynomials. However, SCHRODINGER has shown that in the four-
dimensional space-time serious difficulties appear.

134 THE CONCEPTUAL SITUATION IN PHYSICS

Quite independently from these considerations, the elimination of
the infinities with the help of the factor exp ( D) has been
investigated by PAIS and UHLENBEGK (1950) and others.

The most radical change in the structure of the theory has been
proposed by HEISENBERG (1943). Convinced of the existence of an
absolute length a~io~ 13 cm. or an absolute time r = a\c ~
io" 24 sec, he doubts that the usual description of a physical system
with the help of a HAMILTON function has a meaning at all for
space- and time-intervals smaller than a and r.What we really can
observe are only alterations, in time-intervals long compared with r.
If the state of the system at a time ^ is described by ^(^), that at
t 2 by ^(tfg), it is legitimate to assume that in the equation

the transition operator S(t ly J 2 ) has a physical meaning for t 2 x
> T, in particular its value S( oo, oo). This operator is usually
called the S-matrix. For instance, in a collision process we observe
particles before and after the collision, and we are interested only
to know the distribution after the collision if that before is known.
HEISENBERG maintains that all attempts to describe the collision
process itself should be abandoned.

The postulate of relativistic invariance introduces strange para-
doxes in this theory. The temporal order of events, and thus the
cause-effect relation, breaks down for short time intervals; for
instance, a particle may be absorbed before the creating collision
has taken place. But HEISENBERG (1951) has made it plausible that
these anomalies may be unobservable in principle because of the
atomistic structure of the instruments.

According to the principle of correspondence the S matrix theory
must go over into an ordinary Hamiltonian theory for cases where
the absolute length or time play no important part. HEISENBERG-
comes to the conclusion that very likely the current assumptions
about interactions are not sufficient. These lead to Hamiltonians
which can be re-normalized in the sense described above. Actually
there are indications that a more thorough non-linearity is needed.
In a recent paper (1952) he discusses the process of meson showers-
from this standpoint and uses a type of non-linear field theory which
I found about twenty years ago and published in collaboration with
Infeld (1933, 1934). It is a modification of MAXWELL C S electrodyna-
mics in which the self energy of the electron is finite. MIE had
shown already in 1912 that the equations of the electromagnetic
field can be formally generalized by replacing the linear relations
between the two pairs of field vectors E, B and D, H by non linear

THE CONCEPTUAL SITUATION IN PHYSICS 135

ones. Yet he did not specify these relations, and thus his formalism
remained empty.

The idea which I applied to it is a special case of what WHITTAKER
(1949) has called the principle of impotence. If research leads to an
obstacle which in spite of all efforts cannot be removed, theory
declares it as insurmountable in principle. Well known examples are
the first and second theorems of thermodynamics which are derived
from the impossibility of perpetual motion of the first and second
kind. Other examples are relativity, where the impossibility of
material and signal velocities larger than the velocity of light is
declared, and the uncertainty relations of quantum mechanics,
which forbid the simultaneous determination of position and velocity
and of similar pairs.

In the case of the electromagnetic field the self energy can be
made finite by prohibiting the increase of E the electric vector
beyond a certain limit, the absolute field. This can be done by
imitating relativity where the classical Lagrangian of a free particle
= %mv 2 is replaced by mc 2 [i (i z> 2 /c 2 )*L ^ rom which
v < c follows. In a similar way the Lagrangian density of MAXWELL'S
electrodynamics can be replaced by a square root expression. Thus
a finite self energy of a point charge is obtained which represents
not only the inertial mass but also, as SCHRODINGER has shown, the
gravitational mass.

A more important asset of this theory seems to me the estimate of
the fine structure constant, obtained by HEISENBERG and his pupils
EULER and KOGKEL (1935, 1936) and confirmed by WEISSKOPF
(1936), by comparing the lowest non-linear terms of it with the
corresponding terms of DIRAC'S theory of holes, which are due to
what is called a 'polarization of the vacuum'. The result is i/a
= kje* = 82, which, though still much too small, is of the right
order of magnitude. This method appears to me the only rational
attempt to derive the number i/a = 137.

That the non-linear theory has not found favour is partly due to
the difficulty of quantization, partly to an objection raised by
HEITLER which at the time seemed to me convincing. He said that
a classical theory of the electron, which takes PLANCK'S constant h
as negligible but the charge e as finite, is meaningless because
i/a = hc/e 2 = 137 is a large number.

Now HEISENBERG, in search of a non-linear field theory as
limiting case of his S matrix formalism, took over that square root
method and applied it to the meson field produced by a nucleon.
But he applied it to quite a different type of problem, namely the
meson showers produced by a nuclear collision. Here HEITLER'S

136 THE CONCEPTIONAL SITUATION IN PHYSICS

objection becomes insignificant. If HEISENBERG'S procedure is
analysed, it is seen that it does not rest on the limit h -> o, but
JV - co where JV is the number of quanta involved. In fact BOHR
had both these cases in mind right from the beginning when he
formulated the transition from quantum theory to its classical limit.
(The same consideration justifies the estimate of the fine structure
constant, mentioned above.)

HEISENBERG considers the collision of two nucleons, each being
the source of a meson field, obeying his non-linear field equations.
For a very high collision energy the number of meson quanta will
be very large, hence the application of a classical wave equation
permitted. The total energy carried by this wave i/r can be repre-
sented by an integral over all wave vectors k of a function w(k);
if tt(k) is divided by the energy quantum hv, where v = c \ k | is
the frequency of the wave k, and the result integrated over all k one
obtains the total number JV* of quanta emitted. In this way it can
be shown that for a non-linear theory of the type described multiple
meson production is possible and the value of JV can be estimated.

Now this idea of multiple showers is sharply contradicted, in
particular by HEITLER, who thinks that the observations can be
explained in terms of plural production. The experiments are made
not with two colliding nucleons but with one nucleon hitting a
nucleus; then a cascade of nucleons and mesons will develop and
thus a shower of mesons mixed with nucleons or larger splinters
appear. HEITLER, in a letter to me, quotes experimental investiga-
tions by TERREAUX (1951, 1952) as confirming the cascade theory,
and some unpublished work by McCusKER. Showers were produced
in layers of carbon and of a paraffin containing equal numbers of
G atoms; thus the effect of the H atoms (proton-proton collisions)
can be deduced, and the result was that up to 3 x io 10 eV no
multiple production was observed. This is, however, in strict contra-
diction to experiments made by HAXEL and collaborators, of which
I have learned through my correspondence with HEISENBERG; here
layers of carbon and paraffin of equal mass (equal number of
nucleons) were investigated with the help of counters which recorded
showers of three or more penetrating particles.

The result is that the H atoms have their full share in the multiple
production. HEISENBERG has further sent me a photograph of a
shower containing about 16 mesons, but no heavy track. He inter-
prets it as evidence for multiple production, but it might just as
well be' a nuclear cascade in which the heavy particles are by chance
all neutrons.
Just a few days ago my attention was directed to a paper by

THE CONCEPTIONAL SITUATION IN PHYSICS 137

VIDALE and SCHEIN (1951) which, if confirmed, would settle the
dispute. Self-registering instruments were carried by balloons to
more than 90,000 feet altitude and showers in liquid hydrogen
observed with counters. The results seemed to be in favour of
multiple production, but the assumption made that the primary
particles are nucleons (protons) is not certain at all. I have the
impression that HEISENBERG'S audacious ideas are in the right
direction, and this direction is obviously not backward, but forward
to new abstractions, to a new style of thinking.

I have so far only considered the conceptual problems arising
from the microscopic world of elementary particles. Of equal
importance are the problems of the macrocosmos which are inti-
mately connected with general relativity. However, as I am not an
expert in astrophysics and cosmology, I wish to make only a few
remarks about this vast subject.

Since EDDINGTON'S time we have been aware of the intimate
relation between the atomistic world and the universe. EINSTEIN
himself has made incessant attempts to understand the existence of
particles and quanta as singularities of a united gravitational
electromagnetic field. But I cannot believe that by singling out
these two types of field a real unification can be achieved, quite
apart from my conviction that quantum theory cannot be reduced
to classical concepts. The most important idea, due to astrophysics,
is the suggestion of spontaneous creation of matter. There are two
versions of it, one by HOYLE, BONDI and GOLD (1948), who assume
the permanent creation of hydrogen atoms uniformly in space, the
other by JORDAN (1944), who assumes the instantaneous creation of
whole stars or even galaxies, which then appear as super novae.
Both theories have in common that they oppose the idea of a history
of the universe, as suggested by the simplest interpretation of the
recession of the nebulae (Hubble effect), namely an expanding
universe, beginning, about 2,000 million years ago, in a highly
concentrated state. Instead, both theories aim at describing the
world as being in a steady state, where just as much matter is
created as disappears in infinity (that is when it reaches the velocity
of light).

Both authors have suggested modifications of EINSTEIN'S field
equations. HOYLE'S original theory did not follow the usual
Lagrangian pattern, which secures the compatability of the cause-
effect relation and of general relativity. Thus he, strangely enough,
seemed to be prepared to sacrifice general relativity. McCREA (1951)
has recently shown that this is not necessary, and that by assuming
the existence of a kind of universal cosmic pressure (apart from that

138 THE CONCEPTIONAL SITUATION IN PHYSICS

due to ordinary matter and energy) the relativistic equations can
be preserved.

JORDAN'S theory is based on an idea of DIRAC (1937) according to
which the gravitational constant K is actually not a constant, but a
(slowly changing) eleventh field variable, in addition to the 10
components g^ of the gravitational field. This suggestion is not at
all arbitrary, but based on strong arguments concerning the order
of magnitude of the cosmic constants. JORDAN has further shown
that from the standpoint of group theory his equations are preferable
to those with constant /c, and that the creation of matter in bulk,
as suggested by him, does not mean a violation of the conservation
law of energy, but only a transformation of gravitational energy
into material substance.

Both types of hypotheses are supported by a considerable amount
of empirical evidence which consists, of course, not so much in
direct observations, but in developing a coherent and rational
picture of the universe in agreement with the facts. 1 am unable to
decide who may be nearer to the truth.

I have mentioned these ideas because the future theory of matter
cannot by-pass the cosmological point of view. Very likely I have
omitted to mention other important suggestions, for which I
apologize.

Returning to the first sentences of this lecture, I may say that
much has been achieved during the fifty years since my student days ;
many problems have been solved which about 1900 had not even
been formulated. But the present time seems to offer still more
puzzles, and perhaps harder ones. My aim was to show that our
conceptual armoury will be capable of dealing with them, provided
we do not look back to the good old times, but forward to new
adventures of discovery and explanation.

REFERENCES

BHABHA, H. J. (1945) Rev. Mod. Phys., 17, 200; (1948) Report of 1946 Inter-
national Conference, I: Fundamental Particles (London: The Physical
Society), 22; (1949) Rev. Mod. Phys., 21, 451; (1951) Phys. Rev.,
77, 665.

BLOKHINTZEV, D. (1956) Upsekki fisick nauk, 42, 76; (1951) Ibid., 44, 104.

BOHM, D. (1952) Phys. Rev., 85, 166, 180.

BONDI, H. and GOLD, T. (1948) Mon. Not. Roy. Astr. Soc., 108, 252.

BORN, M. (1933), Nature, Lond., 132, 282; (1934) Proc. Roy. Soc. A, 143,
4*0; ( J 953) && 3- Phil Sci., 4, No. 14, 95.

BORN, M. and GREEN, H. S. (1949) Proc. Roy. Soc. Edinb. A, 62, 470.

BORN, M. and INFELD, L. (1933) Nature, Lond. a 132, 970, 1004; (1934)

Proc. Roy. Soc. A, 144, 425.

THE CONCEPTIONAL SITUATION IN PHYSICS 139

DE BROGUE, L. (1926) C. R. Acad. Sci., Paris, 188, 447; (1927) Ibid., 184,
2 73; l8 5> 380; (1952) see De Broglie, Physicien et Penseur (Paris: A.
Michel).

DIRAC, P. A. M. (1937) Nature, Lord., 139, 323; (1951) Proc. Roy. Soc. A,
209, 291; (1952) Ibid. 9 212, 330.

Du MOND, J. W. M. and COHEN, E. R. (1951) Phys. Rev., 82, 555.

EDDINGTON, A. (1928) Proc. Roy. Soc. A, 121, 524; 122, 358; see also
Relativity Theory of Protons and Electrons (Cambridge: University
Press, 1936). & 7

EULER, H. (1936) Ann. Phys., Lpz., 26, 398.

EULER, H. and HEISENBERG, W. (1936) Phys., 98, 714.

EULER, H. and KOCKEL, B. (1935) Naturwiss., 23, 246.

FIERZ, M. (1939) Helv. Phys. Acta, 12, 3.

FIERZ, M. and PAULI, W. (1939) Proc. Roy. Soc. A, 173, 211.

FRENKEL, J. (1950) Upsekkifaick nauk, 42, 69; (1951) Ibid., 44, no,

HEISENBERG, W. (1943) Phys., 120, 313, 673; (1951) Festschrift Akad. d.
Wiss. Gottingen, p. 50; (1952) . Phys., 133, 65.

HOYLE, F. (1948) Mon. Not. Roy. Astr. Soc., 108, 252.

JORDAN, P. (1944) Phys. ., 45, 183; (1947) Die Herkwft der Sterne (Stutt-
gart: Hirzel); (1949) Nature, Lord., 164, 637; (1952) Schwerkraft
wnd Weltall (Braunschweig: Vieweg).

KARPLUS, R. and KLEIN, A. (1952) Phys. Rev., 85, 972.

KOCKEL, B. (1937) Phys., 107, 153.

LAMB, W. E., Jr. and RETHERFORD, R. C. (1947) Phys. Rev., 72, 241;

XT (1949) Ibid., 75, 1325, 1332; (1950) Ibid., 79> 549-

McCREA, W. H. (1951) Proc. Roy. Soc. A, 206, 562; (1951) J. Trans. Victoria
Inst., 83, 105.

MADELUNG, E. (1927) . Phys., 40, 322.

MIE, G. (1912) Ann. Phys., Lpz., 37, 511; 39, i; (1913) Ibid., 40, i.

MILLER, G. (1951) D. KgL Danske Vidensk. Selskab, Mat-fys. Medd., Nos.
21 and 22.

NEUMANN, J. VON (1932) Mathematische Grundlagen der Quantenmechanik
(Berlin: Springer Verlag), pp. 167-171.

PAIS, A. and UHLENBECK, G. E. (1950) Phys. Rev., 79, 145.

PEIERLS, R. and McMANUs, H. (1948) Proc. Roy. Soc. A, 195, 323.

RAYSKI, J. (1951) Proc. Roy. Soc. A, 206, 575; (1951) Phil. Mag., 42, 1289.

SCHRODINGER, E. (1952) Brit. J. Phil. Sci., 3, 109, 233.

SCHWINGER, J. (1948) Phys. Rev., 74, 1439; (1949) Ibid., 75, 651, 76, 790.

STUCKELBERG, E. C. G. and PETERMANN, A. (1951) Phys. Rev., 82 548;
(1951) Helv. Phys. Acta, 24, 317.

TERREAUX, CH. (1951) Helv. Phys. Acta, 24, 551; (1952) Nuovo Cimento,

9> i29-

TOMANAGA, S. (1946) Prog. Theor. Phys., i, 27, and subsequent papers.
VBDALE, M. L. and SCHEIN, M. (1951) Nuovo Cimento, 8, i.
WEISSKOPF, V. (1936) KgL Danske. Vidensk. Selskab., Mat-fys. Medd., 14, 6.
WEIZSACKER, K. F., VON (1949) Die Geschichte der Natur (Stuttgart: Hirzel);

(1951) gum Weltbild der Physik (Stuttgart: Hirzel).
WHTTTAKER, E. T. (1949) From Euclid to Eddington (Cambridge: University

Press).
YUKAWA, H. (1949) Phys. Rev., 77, 219.

THE INTERPRETATION OF QUANTUM
MECHANICS

[First published in The British Journal for the Philosophy of Science, Vol. IV, 1953.]

following pages are a reply to ERWIN SCHRODINGER'S article,
'Are There Quantum Jumps? Parts I and II', published in
August and November, 1952, in this Journal A discussion on this
subject was to be held in the meeting of the Philosophy of Science
Group on December 8th, 1952, and I was asked to open it. I
accepted this honour rather reluctantly, for I find it awkward to
display in public a disagreement on a fundamental question with
one of my best and oldest friends. Yet I had several motives for
accepting the challenge: The first is my conviction that no dis-
crepancy of opinion on scientific questions can shake our friendship.
The second, that other good and old friends of the same standing as
SCHRODINGER, such as NIELS BOHR, HEiSENBERG and PAUL!, share
my opinion. My third, and the most important reason for entering
into this discussion of SCHRODINGER'S publication is that by its
undeniable literary merits, the width of its historical and philoso-
phical horizon, and the ingenious presentation of the arguments,
it may have a confusing effect on the mind of those who, without
being physicists, are interested in the general ideas of physics.

The discussion on December 8th was rather frustrated by
SCHRODINGER'S absence, due to serious illness. I read my prepared
introduction and answered questions. But this was, of course, not
fair play to SCHRODINGER himself. Therefore I have to state my case
in print. The following is a slightly enlarged version of my intro-
duction to the discussion. As such, it covers not in the least all points
made by SCHRODINGER, but only those which seem to me suited for
a debate amongst philosophers.

i. SCHRODINGER'S CASE RESTATED

The whole discrepancy is not so much an internal matter of
physics, as one of its relation to philosophy and human knowledge
in general. Any one of us theoretical physicists, including SCHROD-
INGER, confronted with an actual problem would use the same, or at
least equivalent mathematical methods, and if we should obtain
concrete results our prediction and our prescription for the experi-
mental verification would be practically the same. The difference of

140

THE INTERPRETATION OF QUANTUM MECHANICS 141

opinion appears only if a philosopher comes along and asks us : Now
what do you really mean by your words, how can you speak about
electrons to be sometimes particles, sometimes waves, and so on?
Such questions about the real meanings of our words are just as
important as the mathematical formalism. SGHRODINGER challenges
the use of words in the current interpretation of the formalism; he
suggests a simple, puristic language and maintains that it can cope
with the situation. We answer, that this purism is not only perfectly
impracticable by its clumsiness, but also quite unjustifiable from the
historical, psychological, epistemological, philosophical standpoint.
I suppose you have all read SCHRODINGER'S paper. What he
maintains can be condensed in a few sentences: The only reality in
the physical world is waves. There are no particles and there are no
energy quanta Av; they are an illusion due to a wrong interpretation
of resonance phenomena of interfering waves. These waves are
connected with integers in a way well known from the vibrations of
strings and other musical instruments, and these integers have
deluded the physicist into believing that they represent numbers of
particles. But there is a special resonance law," characteristic of
quantum mechanics, according to which the sum of the eigenfre-
quencies of two interacting systems remains constant. This has been
interpreted by the physicists as the conservation law of energy
applied to quanta of particles. But there are no such things. Any
attempt to describe the physical phenomena in terms of particles
without contradicting the well-established wave character of their
propagation in space, leads to impossible, unacceptable conceptions
like the assumption of timeless quantum jumps of particles from one
stationary state to another. Moreover, if you try to describe a gas
composed of particles you are compelled to deprive them of their
individuality; if you write tlie symbol (A, B) to express that A is here
at one place, B there at another, the two situations (A, B) and
(B, A) are not only physically indistinguishable, but represent
statistically only one case, not two, as common sense would demand.
All these and many other difficulties disappear if you abandon the
particle concept and use only the idea of waves.

2. ARE THERE ATOMS?

It is only a few years ago since SCHRODINGER published a paper
under the title '2,500 Years of Quantum Mechanics', in which he
stressed the point that PLANCK'S discovery of the quantum was the
culmination of a continuous development starting with the Greek
philosophers LEUQPPUS and DEMOCRTTUS, the founders of the atom-
istic school. At that time he obviously thought the idea that matter is

1^2 THE INTERPRETATION OF QUANTUM MECHANICS

composed of atoms, ultimate indivisible particles, a great achieve-
ment. Now he rejects the same idea, because the execution of the
programme leads to some grinding noise in our logical machinery.

It is this anti-atomistic attitude which appears to me the weakest,
in fact quite indefensible, point in SCHRODINGER'S arguments against
the current interpretation of quantum mechanics. All other points
are of a more technical nature, but this one is fundamental. SCHROD-
INGER opens both parts of his paper by a section entitled The
Cultural Background', in which he accuses the theoretical physicists
of our time of having lost the feeling of historical continuity and
overestimating their own achievements as compared with those of
their forerunners. He gives examples of such defaults which I do not
wish to defend, but I think that he himself offers an example which
is even worse.

The atomistic idea, since its revival through DANIEL BERNOULLI
(1738) in the kinetic theory of gases and through DALTON (1808) in
chemistry, has been so fertile and powerful that SCHRODINGER'S
attempt to overthrow it appears to me almost presumptuous, and in
any case an obvious violation of historical continuity.

3. WAVES INSTEAD OF ATOMS

Such a violation would be justified if he could supply a better and
more powerful substitute. That is exactly what he claims. He says
that everything in physics, and in chemistry as well, can be described
in terms of waves. The ordinary reader will certainly understand this
as meaning: ordinary waves of some not specified substance in
ordinary 3-dimensional space. Only in the last section of Part II
(p. 241) does he indicate that one has in general to do with waves
in a multi-dimensional space, but e To enlarge on this in general terms
would have little value 5 . I think this is a very essential point which
must be discussed. But before doing so I wish to say that I regard
SCHRODINGER'S wave mechanics as one of the most admirable feats in
the whole history of theoretical physics. I also know that his motive
was his dislike of BOHR'S theory of stationary states and quantum
jumps, which he wished to replace by something more reasonable.
I quite understand his triumph when he succeeded in interpreting
those horrible stationary states as innocuous proper vibrations and
the mysterious quantum numbers as the analogy to the numbers of
musical overtones. He is in love with this idea.

I, of course, have no personal attachment to the waves. I have
been involved, together with HEISENBERG and JORDAN, in the
development of another method, matrix mechanics, in which station-
ary states and quantum jumps have a natural place. But I have no

THE INTERPRETATION OF QUANTUM MECHANICS 143

special preference for the matrix theory. As soon as SCHRODINGER'S
wave equation was published, I applied it to the theory of collisions;
this suggested to me the interpretation of the wave function as
probability amplitude. I welcomed SCHRODINGER'S elegant proof of
the formal equivalence of wave mechanics and matrix mechanics.
I do not plead in favour of matrix mechanics, or its generalisation
due to DIRAG, nor do I attack wave mechanics. I wish to refute the
exaggerated claims of SCHRODINGER'S paper from which the non-
expert reader must get the impression that all phenomena can be
described in terms of ordinary waves in ordinary space.

The physicist knows that this is not true. In the case of a 2-body
problem (like the hydrogen atom) one can split the wave equation
into two, one for the motion ofriie centre of mass, the other for the
relative motion, both in 3-dimensional space. But already, in the
case of the 3-body problem (for instance, the helium atom, one
nucleus with two electrons) this is impossible; one needs a 6-dimen-
sional space for the relative motion. In the case of JV particles one
needs a 3 (JV-i) -dimensional space which only in singular cases is
reducible to a smaller number of dimensions.

But this means that the claim of simplicity and of 'Anschaulich-
keit', the possibility of seeing the process in space, is illusory.* In
fact a multi-dimensional wave function is nothing but a name for the
abstract quantity ^ of the formalism, which by some of the modern
theorists also goes under the more learned title of c state vector in the
Hilbert space 5 . Any attempt to describe phenomena, except the
simplest ones, in terms of these multi-dimensional wave functions,
means the formulation of the concise contents of mathematical
formulae in words of ordinary language. This would be not only
extremely clumsy but practically impossible.

In fact, SCHRODINGER makes no attempt in this direction. All his
examples are chosen in such a way that a 3-dimensional representa-
tion is possible. He restricts himself to cases which in the particle
language correspond to independent (non-interacting) particles.
Then he shows that these particles are not behaving as good, well
bred particles, like grains of sand, should behave.

4. WHY ATOMS ARE INDISPENSABLE

I think that in spite of these abnormities the concept of particle
cannot be discarded.

* In another article which has recently appeared ('Louis de Broglie, Physicien et
Penseur' ed. ALBIN MICHEL, Paris, 1952) SCHRSDINGER remarks that the 3-
dimensionality of the waves can be saved with the help of second quantisation.
But the 'Anschaulichkeit* is then also lost and the statistical character of the
v-function is introduced on an even deeper and more abstract level.

144 THE INTERPRETATION OF QUANTUM MECHANICS

As I said already, for the calculations of the theoretical physicist
the whole question is almost irrelevant. But if he wants to connect
his results with experimental facts, he has to describe them in terms
of physical apparatus. These consist of bodies, not of waves. Thus
at some point the wave description, even if it were possible, would
have to be connected with ordinary bodies. The laws governing the
motion of these tangible bodies are undoubtedly those of Newtonian
mechanics. Thus the wave theory has necessarily to provide means
to translate its results into the language of mechanics of ordinary
bodies. If this is done systematically, the connecting link is matrix
mechanics, or one of its generalisations. I cannot see how this transi-
tion from wave mechanics to ordinary mechanics of solid bodies can
possibly be avoided.

Let us look at the matter the other way round, starting from
ordinary bodies. These can be divided into parts, and sub-divided
into still smaller parts. The Greek idea was that this procedure has
an end somewhere, when parts become particles, atoms, which are
indivisible.

Modern theory has modified this view to some degree, but I need
not go into details which you all know. The parts of a substance
obtained by division and subdivision are of the same physical nature
until you approach the chemical atom. This is not indivisible, but its
parts are of a different nature, particles of a more subtle quality,
nucleons and electrons. Then we discover that the smallest units,
the chemical. atoms and still more the nucleons and electrons, have
not only different qualities, but decidedly strange qualities, strange
if you expect always to find the same as you are accustomed to. They
behave differently from the powder particles into which you have
first ground your material. They have no individuality, their
position and velocity can be determined only with a restricted
accuracy (according to HEISENBERG'S uncertainty relation) and so
on. Shall we then say, well, there are no particles any more, we must
regretfully abandon the use of this simple and attractive picture ?

We can do it if we take a strictly positivistic standpoint: The
only reality is the sense impressions. All the rest are 'constructs' of
the mind. We are able to predict, with the help of the mathematical
apparatus of quantum mechanics, what the experimentalist will
observe under definite experimental conditions, the current shown
by a galvanometer, the track in a photographic plate. But it is
meaningless to ask what there is behind the phenomena, waves or
particles or what else. Many physicists have adopted this standpoint.
I dislike it thoroughly, and so does SGHRODINGER. For he insists that
there is something behind the phenomena, the sense impressions,

THE INTERPRETATION OF QUANTUM MECHANICS 45

namely waves moving in a still scantily explored medium. Recently
an American physicist, BOHM, has taken the opposite standpoint; he
claims that he can interpret the whole of quantum mechanics in
terms of ordinary particles with the help of parameters describing
unobservable 'concealed' processes.

5. HOW TO MODIFY THE ATOMISTIC CONCEPT

I think' that neither of these extremist views can be maintained.
The current interpretation of quantum theory which tries to recon-
cile both aspects of the phenomena, waves, and particles, seems to
me on the right way. It is impossible to give here an account of the
intricate logical balance. I wish only to illustrate the manner in
which the particle concept is adapted to new conditions, by some
examples from other fields, where a similar situation is found. It is
of course no new situation that a concept in its original meaning
turns out too narrow. But instead of abandoning it, science has
applied another method, which is by far more fertile and satisfac-
tory. Consider the example of the number concept. Number means
originally what we now call integer, i, 2, 3 ... KRONEKER has said
that God has made the integers, while the rest are human work.
Indeed, if you define numbers as the means of counting things, even
rational numbers like 2/3 or 4/5 are not numbers any more. The
Greeks extended the concept of number to them by restricting the
consideration to a finite set where a smallest unit (the greatest
common denominator) can be found. But then they made the
fundamental discovery that the diagonal of the square (of the side i),
which we write y's, is not a number in this sense; but great as their
logical genius was, they did not make the next constructive step.
They had not the pluck to generalise the number concept in such a
way that <\/2 was included, but invented an ingenious yet rather
clumsy geometrical method to deal with such cases. This was the
stumbling block which retarded mathematics for about 2,000 years.
Only in modern times the necessary generalisation of the idea of
number was made so as to include these things such as *y/2, still
called irrational. But then further generalisations followed, the
introduction of algebraic, transcendental, complex numbers. You
cannot count with the help of these. But they have other, more
formal properties in common with the integers, and the latter are a
special case. Similar generalisations of concepts are common in
mathematics. But they appear also in physics. Sound was certainly
defined as that which you can hear, light as that which you can see.
But we speak now of inaudible sound (ultrasonics) and invisible light
(infrared, ultraviolet). Even in ordinary life this process of extension

146 THE INTERPRETATION OF QUANTUM MECHANICS

of meaning is going on. Take the concept of democracy which
originally meant the organisation of government in the Greek city
states where the citizens assembled in the market place to discuss
and decide their problems; today, it is used for the government of
gigantic states by parliamentary representation. In Russia it even
means something which we should regard as the opposite of
democracy. Therefore we had better return to the safe ground of
science.

I maintain that the use of the concept of particles has to be
justified in the same way. It must satisfy two conditions: First it must
share some (not in the least all) properties of the primitive idea of
particle (to be part of matter in bulk, of which it can be regarded
as composed), and secondly this primitive idea must be a special,
or better,, limiting case.

Now it is exactly in this sense that the particle concept is used in
quantum mechanics. I cannot see any objection to it. SCHRODINGER'S
examples seem to me of the kind which prohibited the Greeks from
admitting the representation of the diagonal of the unit square as a
number; it differs from all possible ratios of integers, as can easily be
seen. The effect of accepting SCHRODINGER'S thesis would perhaps not
be equally portentous, because he does not attack the formal theory,
only its philosophical background. He would even allow the
physicists and chemists to use the particle language with a proper
e as if. Imagine a textbook of chemistry written according to this
prescription. Water behaves as if it were composed of molecules
H 2 O, each of which again reacts as if composed of two H-atoms and
one O-atom. But when we continue, each H-atom has properties
as if it were composed of a nucleus and an electron, we transgress the
permitted domain of c as if', for here SCHRODINGER insists that there is
no particle called electron but a charged wave around the nucleus
which itself actually is also a wave of some kind. But when we then
wish to deal with a photo-ionisation of this H-atom we have to fall
back to his *as if 3 to describe the discontinuous recording of a Geiger
counter.

All our language, in life and science, is growing through general-
isations of concepts, which sometimes are first considered to be e as
ifs% but then are amalgamated and become legitimate words in
their own right. For this end it is necessary to fix the rules of their
employment in a reasonable manner. This process, in which NIELS
BOHR has played a leading part, is still going on, and, I think, with
fair success. One can, of course, pick out points where some logical
hardness or roughness appears, and that is what SCHRODINGER has
done.

THE INTERPRETATION OF QUANTUM MECHANICS 147

On the other hand, SCHRODINGER cannot avoid the use of the
words particles or atoms. They appear in many of his examples;
otherwise his words would convey no meaning. For instance, when
he speaks about quantum statistics of gases he has to discuss a wave
equation in a multi-dimensional space. This equation has, of course,
a simple meaning if considered from the particle standpoint ; it is the
wave-mechanical translation of the law of conservation of kinetic
energy for n particles. Now SCHRODINGER is compelled to disown
this translation, the lovely child of his brain, for otherwise he would
admit that there are, in some sense, particles. He has to take the 3/2-
dimensional wave equation as something given to him by inspiration
and confirmed by experiments. This is a distortion of historical facts.

6. COLLISIONS

Though I wish to avoid technical details I have to say a few words
about the problem of collisions which SCHRODINGER discusses in
several places (Sections 6 and 8). He finds the usual quantum-
mechanical treatment faulty, he accuses the physicists of loose
speech, he preaches to them that 'Science is not a soliloquy' and
prophesies that their work will be forgotten in 2 3 ooo years' time,
while that of ARCHIMEDES or GALILEO has survived similar periods.
In a letter to me he maintains that 'almost all great successes of
quantum mechanics consist of the satisfactory calculation of ex-
tended systems of eigenvalues (of the energy), each from a definite,
more or less plausible assumption about the nature of the system in
question (Hamilton operator), and have nothing at all to do with the
statistical interpretation. On the other side there are the scattering
experiments (calculation of differential cross-sections of interaction
and things like that) . Only the Klein-Nishina formula is apparently
quantitatively confirmed. (The latter represents the scattering of
light, or photons, by an electron.) 5 He further doubts that the
statistical interpretation, which I have first suggested and which has
been formulated in the most general way by VON NEUMANN, is
applicable to these cases at all.

To this I reply that in principle we know about the eigenvalues
of the energy (Hamiltonian) of material systems only from experi-
ments about emission, absorption, scattering of light or electrons.
These processes are all due to the coupling of the system considered
with a 'messenger' field (the electromagnetic or photon field, or DE
BROGLIE'S electron field) and it seems to me quite arbitrary to pick
out the scattering as less reputable than the other two effects.
Further, a look into the literature, for instance, the well-known book
by MOTT and MASSEY, or the important articles by NIELS BOHR, on

148 THE INTERPRETATION Of QUANTUM MECHANICS

the penetration of particles through matter and innumerable other
papers and books, shows that the number of more or less quantitative
confirmations of the quantum-statistical scattering laws is very large,
and that there are qualitative confirmations of a particularly con-
vincing kind. Even in nuclear physics, where the knowledge of the
interaction law (Hamiltonian) is doubtful and scanty, the principles
of the statistical theory have been used with great success, of which
the atomic bomb is one very impressive example.

Concerning SCHRODINGER'S scepticism about the applicability of
the general scheme for transitions (quantum jumps) to the case of
collisions I am unable to follow his reasoning. He describes the
procedure as if a collision were a transition between two states of
different energy. In fact the typical 'elastic 9 collision is a transition
between states of equal energy but different momentum vectors.
My original method dealing with this case avoids any reference to
time; it considers the steady state of an incoming wave (representing
a beam of e messenger' particles), transformed by its interaction with
an atom into a spherical wave (representing the out-going, scattered
particles). In this way of considering the process there is no initial
and no final state, concepts which seem to SCHRODINGER ill-defined.
They appear in DIRAG'S version of the collision theory which he
developed in order to consider collisions as a special case of the
general theory of transitions in time (formulated first in my papers on
'adiabatic invariants' and in DIRAG'S simultaneous publications, and
perfected by J. v. NEUMANN). But DIRAG has shown that his method
(involving time) is mathematically equivalent to the 'stationary,
method; the conceptual difficulties which worry SCHRODINGER are
therefore only a matter of careful formulation.

Another objection which he raises refers to the approximation
method which I introduced in my early papers to solve the very
complicated mathematical equations of scattering. This method
gives reasonable, and often well-confirmed, results in the first
approximation; but higher approximations are difficult to obtain,
and if they can be constructed there are cases where they lead to
divergent integrals. However, there are other methods which use
quite different expansions (for instance, in terms of spherical
harmonics and Bessel functions) and lead to results which are
mathematically sound and well confirmed by experiments.

I cannot see at all that these purely mathematical objections have
anything to do with the question of 'particles-waves', or 'quantum
jumps'. For if we accept SGHRODINGER'S standpoint that there are
no particles, only waves, the scattering calculations would be
exactly the same as before; the only difference would be that we

THE INTERPRETATION o QUANTUM MECHANICS 149

would speak about the intensity of the incoming and the outgoing
wave (electromagnetic, electronic, pro tonic, etc., wave, as the case
may be), and omit to interpret this intensity as the probability of
the appearance of particles. The real problem raised by SCHROD-
INGER is whether this probability interpretation is significant. His
mathematical scruples have nothing to do with it. To decide this
significant question, consider, for instance, RUTHERFORD'S experi-
ments about the scattering of a-rays by nuclei. Here, by a kind
of lucky mathematical co-incidence, the classical calculation (using
particles obeying the laws of Newtonian mechanics) and the wave-
mechanical calculation (which can be performed rigorously in this
case) give the same result. This result is confirmed by counting the
a-particles in the incoming and in the outgoing beam (for different
directions of scattering) . The result is completely independent of
the method of counting, whether by scintillations of a zinc-sulphide
screen, or by different types of counters. How does SCHRODINGER
account for this fact? As far as I see he has no ready explanation.
He seems to think that it is not a discontinuity in the beam which
produces the countable events, but some feature of the counting
instrument. But how then is it to be explained that the result is
independent of the type of instrument, even to that degree, that
sparks in the little crystals of the zinc-sulphide screen and gas
tubes, connected with elaborate amplifier apparatus, count the
same (average) number of events? Here SGHRODINGER'S bias against
the particle idea leads him to an almost mystical attitude; he hopes
that the future will solve this riddle in a satisfactory way.

7. CONCLUSION

I have refrained from discussing the statistical interpretation of
quantum mechanics in detail. This is not a simple matter, and
demands not only the knowledge of a complicated mathematical
formalism, but a certain philosophical attitude: the willingness to
sacrifice traditional concepts and to accept new ones, like BOHR'S
principle of complementarity. I am far from saying that the present
interpretation is perfect and final. I welcome SCHRODINGER'S attack
against the complacency of many physicists who are accepting the
current interpretation because it works, without worrying about the
soundness of the foundations. Yet I do not think that SCHRODINGER
has made a positive contribution to the philosophical problems. It
is very awkward for me to criticise the philosophy of a friend whom
I deeply admire as a great scholar and deep thinker. Therefore I
shall make use of a method of defence which SCHRODINGER himself
is not too proud to use, namely the quotation of authorities who share

I5O THE INTERPRETATION OF QUANTUM MECHANICS

my own opinion. I choose as my witness W. PAULI who is generally
acknowledged to be the most critical, logically and mathematically
exacting amongst the scholars who have contributed to quantum
mechanics. I translate a few lines from a letter (in German) which
I have recently received:

Against all retrograde efforts (SCHRODINGER, BOHM, etc., and
in a certain sense, also EINSTEIN) I am certain that the statistical
character of the ^-function, and thus of the laws of nature
which you have, right from the beginning, strongly stressed in
opposition to SCHRODINGER will determine the style of the
laws for at least some centuries. It is possible that later, for
example in connection with the processes of life, something
entirely new may be found, but to dream of a way back, back
to the classical style of NEWTON-MAXWELL (and it is nothing
but dreams which those gentlemen indulge in), that seems to
me hopeless, off the way, bad taste. And we could add c it is
not even a lovely dream'.

What PAULI means by the 'style' of a conceptual structure you
might prefer to call the philosophical attitude of a period, which
determines the cultural background. It is here that we differ, and
the auspices of an agreement are therefore frail.

PHYSICAL REALITY

[First published in Philosophical Quarterly, pp. 139-149, 1953.]

notion of reality in the physical world has become, during
the last century, somewhat problematic. The contrast between
the simple and obvious reality of the innumerable instruments,
machines, engines, and gadgets produced by our technological
industry, which is applied physics, and of the vague and abstract
reality of the fundamental concepts of physical science, as forces
and fields, particles and quanta, is doubtless bewildering. There has
already developed a gap between pure and applied science and
between the groups of men devoted to the one or the other activity,
a separation which may lead to a dangerous estrangement. Physics
needs a unifying philosophy, expressible in ordinary language, to
bridge this gulf between 'reality' as thought of in practice and in
theory. I am not a philosopher but a theoretical physicist. I cannot
provide a well balanced philosophy of science that would take due
account of the ideas developed by differing schools, but I shall
endeavour to formulate some ideas which have helped me in my
own struggle with these problems.

There is a school of thought amongst theoretical physicists and
scientific philosophers which advocates a standpoint radically
abstract. This philosophy was expressed, for instance, in the notable
lecture given by Professor H. DINGLE to Section A of the British
Association in Edinburgh (published in Nature, 168, 1951, p. 630)
and I cannot explain my own standpoint better than by way of
contrast. But in quoting extracts from DINGLE'S lecture I do not
intend to conduct a personal controversy; these quotations serve
only as examples suitable to develop my own differing views. Let us
begin with the following sentence: 'The quantities with which
physics concerns itself are not evaluations of objective properties of
parts of the external material world; they are simply the results we
obtain when we perform certain operations. 5 This looks like a denial
of the existence of a pre-existing material world; it suggests that the
physicist does not care about the real world and makes an experi-
ment solely in order to predict the results of yet another experiment.
Why the physicist should take the trouble to make an experiment at
all is not explained. This question is seemingly regarded as not
worthy of a philosopher of science. Can we avoid asking what is the
part played in this scheme of things by the instruments, made of

152 PHYSICAL REALITY

steel, brass, glass, etc., carefully composed and adjusted for an
experiment? Are they, too, no part of a pre-existing external material
world? Are they, like electrons, atoms and fields, merely abstract
ideas used to predict the phenomena to be observed at the next
experiment which is again only an assembly of ghosts ? We have
before us a standpoint of extreme subjectivism, which may rightly
be called 'physical solipsism'. It is well-known that obstinately held
solipsism cannot be refuted by logical argument. This much, how-
ever, can be said, that solipsism such as this does not solve but
evades the problem. Logical coherence is a purely negative criterion;
no system can be accepted without it, but no system is acceptable
just because it is logically tenable. The only positive argument in
support of this abstract type of ultra-subjectivism is an historical
one. It is maintained that the belief in the existence of an external
world is irrelevant and indeed detrimental to the progress of science,
and that what the physicist is doing can be satisfactorily understood
only in terms of Experiences', not of the external world.

The actual situation is very different. All great discoveries in
experimental physics have been due to the intuition of men who
made free use of models, which were for them not products of the
imagination, but representatives of real things. How could an
experimentalist work and communicate with his collaborators and
his contemporaries without using models composed of particles,
electrons, nucleons, photons, neutrinos, fields and waves, the con-
cepts of which are condemned as irrelevant and futile ?

However, there is of course some reason for this extreme stand-
point. We have learned that a certain caution is necessary in using
these concepts. The naive approach to the problem of reality which
was so successful in the classical or Newtonian period, has been
proved to be not satisfactory. Modern theories demand a reformula-
tion. This new formulation is slowly evolving, but has probably not
reached a final expression. I shall try to indicate the present
tendencies.

The first point is to remember that the word reality is part of our
ordinary language, and hence its meaning is ambiguous like that of
most words. There are subjective philosophies which teach that only
the mental world is real and the physical world merely an appear-
ance, a shadow without substance. This standpoint, though of the
greatest philosophical interest, is outside the scope of our discussion,
which has to do only with physical reality. Still there remain enough
other queries. The realities of a peasant or craftsman, a merchant
or banker, a statesman or soldier have certainly little in common.
For each of these the most real things are those which occupy the

PHYSICAL REALITY 153

centre of his mind, the word real being used as almost synonymous
with important. I wonder whether any philosophy can give a
definition of the concept of reality that is untainted by some such
subjective associations. The question concerning us is whether
science can.

This leads to the second point, stressed by DINGLE, whether the
use of the concept and word 'reality* can be discarded without
detriment to science. My answer is that it could only be disregarded
by men isolated in ivory towers, remote from all experience, from
all actual doing and observing, the type of man who becomes
extremely absorbed in pure mathematics, metaphysics or logic.
NIELS BOHR, who has contributed more to the philosophy of modern
science than anybody else, has repeatedly and emphatically said
that it is impossible to describe any actual experiment without using
ordinary language and the concepts of naive realism. Without this
concession no communication about facts is conceivable, even
between the most sublime minds. And it is an essential part of this
procedure to distinguish between ideas, projects, theories and
formulae on the one side, and the real instruments and gadgets
constructed according to those ideas. Here the naive use of the
word real, the simple belief in the real existence of the material
apparatus, is imperative. I presume that the abstract school repre-
sented by DINGLE does not deny this, although he does not say so.
He does, however, forbid the application of the concept of reality
to atoms, electrons, fields, etc., terms used in the interpretation of
observations. But where is the border between these two domains ?
Start with a piece of a crystal, which belongs to the domain of
crude reality, and grind it into a powder, whose particles are too
small to be seen by the unaided eye. You have to take a microscope:
Are the particles then less real? Still smaller particles, colloids,
appear, properly illuminated, in the ultra-microscope, as bright
points without structure. There is a continuous transition between
these particles and single molecules or atoms. The ultra-microscope
there deserts you. You then have the electron microscope with
which you can see even large molecules. Where does that crude
reality, in which the experimentalist lives, end, and where does the
atomistic world, in which the idea of reality is illusion and anathema,
begin?

There is, of course, no such border; if we are compelled to
attribute reality to the ordinary things of everyday life including
scientific instruments and materials used in experimenting, we cannot
cease doing so for objects observable only with the help of instru-
ments. To call these objects real and part of the external world

154 PHYSICAL REALITY

does not, however, commit us in any way to any definite description:
a thing may be real though very different from other things we know.
Let me now discuss some examples which DINGLE cites to show
the failure in physics of the concept of an objective reality.

The first example is the kinetic theory of matter. DINGLE discusses
the statistical method, which is not concerned with the single orbits
of the molecules and is content to calculate averages, in order to
represent 'observations (that is, appearances)' and he calls this
attitude a 'betrayal of the true mission of physics according to the
accepted philosophy. They (the physicists) were dedicated to the
investigation of reality, which had become the investigation of the
nature and behaviour of molecules; and instead of pursuing that,
they occupied themselves in showing how their ignorance of reality
could be used in order to describe mere appearances'. I have not
been able to understand whether DINGLE thinks the whole kinetic
theory superfluous, or whether he suggests stripping the molecules
of their reality by calling them 'counters' or 'dummies'. For he
makes no attempt to analyse the actual evidence provided by the
kinetic theory for the existence of molecules. Let me sketch such an
analysis in a few words.

The kinetic derivation of BOYLE'S law establishes only the pos-
sibility of an atomistic explanation, and can hardly be called
evidence. However, the same derivation properly formulated leads
to a definite value of the mean energy, hence of the specific heat (f R
for monatomic gases, R being the gas constant) which no pheno-
menological consideration could provide. The general formula for
the mean energy contains the numbers of degrees of freedom of the
molecules or 'dummies', to use DINGLE'S expression. The kinetic
interpretation of the deviations from BOYLE'S laws leads to an
estimate of the size of the molecules, which is confirmed by a quite
different set of phenomena, the irreversible processes of heat conduc-
tion, viscosity, diffusion. Many concepts first introduced in a
theoretical way, like velocity distribution, free path, etc., have been
confirmed and determined by direct measurements. The fluctua-
tions predicted by the kinetic theory are observable in many ways,
through the Brownian motion, the blue colour of the sky, etc. Of
course, as DINGLE says, these are all phenomena, 'appearances', the
molecules remaining in the background. But the essential point, not
mentioned by DINGLE, is that the kinetic theory leads to definite
properties of the molecules, weight, size, shape (degrees of freedom),
mutual interaction. A small number of molecular constants de-
termines an unlimited number of phenomenological properties, in
virtue of the molecular hypothesis. Therefore each new property is

PHYSICAL REALITY 155

a confirmation of the molecular hypothesis. Amongst these predic-
tions are such amazing feats as VON LAUE'S X-ray patterns produced
by crystals, and the whole range of radioactive phenomena. Here
the evidence of the reality of molecules is striking indeed, and to
speak of a 'dummy' producing a track in a Wilson chamber or a
photographic emulsion seems to me to say the least inadequate.
Compare this kind of reality with the following example: You see a
gun fired and, a hundred yards away 5 a man breaking down. How
do you know that the bullet sticking in the man's wound has actually
flown from the gun to the body? Nobody has seen it, in fact nobody
could have seen it, except a scientist after cumbersome preparations,
e.g. through the installation of a complicated optical apparatus of
the kind ERNST MACH invented for photographing flying projectiles.
Yet I am sure you believe that the bullet has in the short interval
between the firing of the gun and the wounding of the man, per-
formed a definite trajectory; you believe that it was really there
during the interval; or are you content to say, Oh, I don't know;
it's enough to know the phenomena of the firing and wounding. All
things between are theoretical imagination, the bullet in flight is
merely a "dummy" invented to account for the connection of the
two phenomena by the laws of mechanics'. I cannot refute this
attitude by logical reasoning. I only wish to point out that if one
denies the existential evidence of an atomic track which can be seen,
one is committed to denying the existence of a bullet in flight which
cannot be seen, and of numerous similar things.

The root of this strange denial of reality to things like molecules is
the interpretation of the concept 'real' as meaning 'known in every
detail 9 . This does not agree with the usual application of the word.
We think all the 500 millions of Chinese are real, although we know
not a single one, or perhaps a few individuals, and have not the
slightest knowledge of their whereabouts, activities, motions, reac-
tions. We think the Romans of Caesar's time or the Chinese during
the life of CONFUCIUS were real although we have no possible means
of verifying this in the way which" DINGLE demands in the case of
molecules. Are these Romans or Chinese of the present or the past
only dummies invented by the historians to connect phenomena?
Which phenomena? Perhaps the words found in newspapers, in
books, or on ancient tombstones?

All these considerations are rather on the surface and do not
touch the actual difficulties which physics encounters, and which
compel us to revise our fundamental notions. DINGLE'S next example,
relativity, leads a little nearer to these problems. He asserts that in
accordance with the philosophy of the time, the real material world,

156 PHVSICAL REALITY

whether regarded as consisting of molecules or of gross bodies, was
conceived to possess its properties by intrinsic right. Thus its con-
stituents had a size, a mass, a velocity, and so on'. After elaborating
this he continues: 'Now the basic requirement of the theory of
relativity was that all these properties were almost completely
indefinite', and he exemplifies this by the notions of length and of
mass, which according to relativity depend on the velocity of the
observer. The same distance measured by different observers in
relative motion may be anything between a maximum and nothing,
the same mass anything between a minimum and infinity. He con-
cludes that by abandoning all attempts to assign any property at
all to matter we can learn more and more about the relations of
phenomena*. Now this is a misrepresentation of the theory of
relativity, which has never abandoned all attempts to assign
properties to matter, but has refined the method of doing so in order
to conform with certain new experiences, such as the famous
Michelson-Morley experiment.

In fact this example is very well suited to get at the root of the
matter. This root of the matter is a very simple logical distinction
which seems to be obvious to anybody not biased by a solipsistic
metaphysics; namely this: that often a measurable quantity is not a
thing, but a property of its relation to other things. To give an
example: Gut out a figure, say a circle, of a piece of cardboard and
observe its shadow thrown by a distant lamp on a plane wall. The
shadow of the circle will appear in general as an ellipse, and by
turning your cardboard figure you can give to the length of an
axis of the elliptical shadow any value between almost zero and a
maximum. That is the exact analogue of the behaviour of length in
relativity which in different states of motion may have any value
between zero and a maximum. If you wish to have an analogue to
the behaviour of mass which according to velocity may have any
value between a minimum and infinity, take a long sausage and cut
slices with different inclination which will be ellipses with one axis
between a minimum and 'practical' infinity. To return to the
shadow of the circle, it is evident that the simultaneous observation
of the shadows on several different planes suffices to ascertain the
fact that the original cardboard figure is a circle and to determine
uniquely its radius. This radius is what mathematicians call an
invariant for the transformations produced by parallel projection. In
the same way there is an invariant of all the cross sections of a
sausage, that with the smallest area. Most measurements in physics
are not directly concerned with the things which interest us, but
with some kind of projection, this word taken in the widest possible

PHYSICAL REALITY 157

sense. The expression co-ordinate or component can also be so used.

The projection (the shadow in our example) is defined in relation
to a system of reference (the walls, on which the shadow may be
thrown). There are in general many equivalent systems of reference.
In every physical theory there is a rule which connects the projections
of the same object on different systems of reference, called a law of
transformation, and all these transformations have the property of
forming a group, i.e. the sequence of two consecutive transforma-
tions is a transformation of the same kind. Invariants are quantities
having the same value for any systems of reference, hence they are
independent of the transformations.

Now the main advances in the conceptual structure of physics
consist in the discovery that some quantity which was regarded as
the property of a thing is in fact only the property of a projection.

The development of the theory of gravity is an example. Using
modern mathematical language, the primitive (pre-Newtonian)
conception of gravity is connected with a group of transformations
for which the vertical, the normal to the plane surface of the earth,
is absolutely fixed. For these transformations the size and direction
of the force of gravity is an invariant which implies that the weight
is an intrinsic property of the body which it carries along. The
situation changed completely when NEWTON discovered gravity to be
a special case of general gravitation. The group of transformations
was extended in such a way that space became isotropic, with no
fixed direction; gravity then became just a component of the
gravitational force.

The theory of relativity has continued this development. The
transformations of classical mechanics, often called Galilean trans-
formations, kept space and time apart. The experiences condensed
in the theory of relativity showed that this does not agree with facts.
One has to use a wider group, called Lorentz transformations, in
order to introduce an intimate connection between space co-ordinates
and time. Naturally, quantities regarded by the older theory as
invariants, like distances in rigid systems, time intervals shown by
clocks in different positions, masses of bodies, are now found to be
projections, components of invariant quantities not directly acces-
sible. Still, as in the case of the shadow, by deterniining a number of
these components, the invariants can be found. Thus it turns out
that the maximum length and the minimum mass are relativistic
invariants. It would perhaps have been preferable to call these
invariants, which are properties of bodies, by the old names length,
time, mass, and to invent new names for the projections. But science
is strangely conservative in such matters, and it has been agreed to

158 PHYSICAL REALITY

rename the invariants rest-length, proper- time, rest-mass etc., and
keep the old expressions for the components, although these are
now not properties of a body but of its relation to a system of
reference.

I think the idea of invariant is the clue to a rational concept of
reality, not only in physics but in every aspect of the world.

The theory of transformation groups and their invariants is a well-
established part of mathematics. Already in 1872 the great mathe-
matician FELIX KLEIN discussed in his famous 'Erlanger Programm'
the classification of geometry according to this point of view; the
theory of relativity can be regarded as an extension of this programme
to the four-dimensional geometry of space-time. The question of
reality in regard to gross matter has from this standpoint a clear
and simple answer.

The situation is more difficult in atomic physics. It is well known
that the laws of quantum mechanics lead to a kind of indeterminacy
expressed by HEISENBERG'S uncertainty relations. Is not this vague-
ness, this impossibility of answering definite questions about position
and velocity of a particle, an argument against the reality of particles
and altogether of the objective, real world ? Here we have to reflect
about what we mean by a particle, for instance a photon, an
electron, a meson, a nucleon in regard to the experimental evidence;
and again we find that these words signify definite invariants which
can be unambiguously constructed by combining a number of
observations.

The underlying transformation theory, however, is rather involved,
and I can give here only a short, sketchy indication. The essence of
the matter can be explained with the help of ordinary light.

The wave character of light was established by YOUNG and
FRESNEL by showing that two beams of light, produced by splitting
one beam, when re-united give interference fringes. Almost a
hundred years later EINSTEIN interpreted the photo-electric effect as
the action of light quanta or photons which on hitting a metal
surface knock out electrons. Thus light has in addition a corpuscular
aspect, a fact confirmed by innumerable experiments. The strange
thing is that between these apparently contradictory concepts there
exists a simple quantitative relation, which PLANCK had derived
already five years earlier from the behaviour of heat radiation,
namely E = Av, where E is the energy of the photon, v the frequency
of the wave, and h a constant. The conceptual difficulty comes from
the fact that the energy E is concentrated in a very small particle
while the frequency v, or better the wave length A = c\v, needs for
definition a (practically) infinite train of waves.

PHYSICAL REALITY 159

This paradox can only be solved by sacrificing some traditional
concept. As we now know, what we have to give up is the idea that
the particles, considered by themselves, follow deterministic laws
similar to those of classical mechanics. The theory can predict only
probabilities, and these are determined by the waves (they are the
squares of the amplitudes). This is of course a decisive change in our
attitude to nature. It calls for new ways of describing the physical
world, but not the denial of its reality. The essence of the new
method can be seen from a simple example.

Let a beam of light pass through a Nicol prism; it thus becomes
linearly polarised. Let this primary beam, which may have the
amplitude A, pass through a double-refractory crystal; there emerge
two secondary beams, linearly polarised perpendicularly to one
another. If is the angle between the direction of polarisation of
the primary and of one of the secondary beams, the amplitudes of
the latter are A cos 6 and A sin d. Their intensities are therefore in
the ratio cos 2 6 : sin 2 6. If now the primary intensity is decreased
until you see nothing with your eyes, you still can observe the
arrival of photons with the help of a sensitive photocell and of proper
amplification, and you can count the number of photons. Thus you
will find that their average number in the two secondary beams is
in the ratio of cos 2 6 : sin 2 6. This is the simplest example of the
statistical interpretation mentioned above, that probabilities are
determined by the squares of the amplitudes of the waves. The point
to which I wish to direct attention is that these secondary amplitudes
are the projections of the primary amplitude in two directions
determined by the instrument. The prediction made by the theory in
regard to the intensities of the emerging beams, or the number of
photons in these, has a meaning only in relation to the whole
experimental arrangement, the Nicol prism and the crystal.

Now this example is typical for quantum phenomena. Take for
example the corresponding experiment with electrons, known as the
Stern-Gerlach effect, where the Nicol prism is replaced by a non-
homogeneous magnetic field and the polarization by the direction
of the spin. Again the observable part, the number of electrons of a
given spin, depends on the special experimental arrangement in a
way which can be described by saying that the instrument records
projections of the actual state.

This description applies to any quantum effect. An observation or
measurement does not refer to a natural phenomenon as such, but
to its aspect from, or its projection on, a system of reference which
as a matter of fact is the whole apparatus used. Expressed in
mathematical terms the word projection is perfectly justified since

l6o PHYSICAL REALITY

the main operation is a direct generalization of the geometrical act
of projecting, only in a space of many, often infinitely many,
dimensions.

If these facts are analysed from the standpoint of particles alone,
there appear those uncertainty relations, which I shall not discuss
here, since they are now to be found in every textbook of quantum
mechanics. BOHR has introduced the idea of complementarity to
express the fact that the maximum knowledge of a physical entity
cannot be obtained from a single observation or a single experimental
arrangement, but that different experimental arrangements,
mutually exclusive but complementary, are necessary. In the
language proposed here this would mean that the maximum know-
ledge can only be obtained by a sufficient number of independent
projections of the same physical entity, just as in the case of the
circular piece of cardboard, where the shadows on several planes
were necessary to determine its shape and invariant (radius). The
observations of the different shadows on two perpendicular planes,
used above to explain the concept of the invariant, also illustrate
very well the essence of the idea of complementarity. The final
result of complementary experiments is a set of invariants, charac-
teristic of the entity. The main invariants are called charge, mass (or
rather: rest-mass), spin, etc.; and in every instance, when we are
able to determine these quantities, we decide we have to do with a
definite particle. I maintain that we are justified in regarding these
particles as real in a sense not essentially different from the usual
meaning of the word.

Before defending this standpoint I wish to discuss in a few words
the remark often repeated that quantum mechanics has destroyed
the distinction between object and subject, since it cannot describe
a situation in nature as such, but only that produced by a man-
made experiment. This is perfectly true. The atomic physicist is
very far removed from the idyllic attitude of the old-fashioned
naturalist who, by watching butterflies in a meadow, hoped to
penetrate into Nature's mysteries. The observation of atomic
phenomena needs instruments of such sensitivity that their reaction
in making measurements must be taken into account, and, as this
reaction is subject to the same quantum laws as the particles ob-
served, a degree of uncertainty is introduced, which prohibits
deterministic prediction. It is therefore obviously futile to ponder
about the situation which would have arisen without the inter-
ference of the observer, or independent of the observer. But in
respect to a given interference of the observer, in a given experi-
mental situation, quantum mechanics makes definite statements as

PHYSICAL REALITY l6l

to the maximum information obtainable. Although we cannot know
everything, nor even approximate to a knowledge which is complete,
by improving our instruments we can obtain certain restricted 3 but
well described, information which is independent of the observer
and his apparatus, namely the invariant features of a number of
properly devised experiments. The process of acquiring this informa-
tion is certainly conditioned by the subject observing; but that does
not mean that the results lack reality. For obviously the experi-
mentalist with his apparatus is part of the real world, and even the
mental processes used in designing his experiment are real. The
boundary between the action of the subject and the reaction of the
object is blurred indeed. But this does not prohibit us from using
these concepts in a reasonable way. The boundary of a liquid and
its vapour is also blurred, as their atoms are permanently evapora-
ting and condensing. Still we can speak of liquid and vapour.

Let us now return to the question of reality and recall the views
of some modern philosophers on the subject.

In a recent book the American writer, H. MARGENAU, advocates
the standpoint that reality consists of two layers: the immediate
data of the senses, and 'constructs'; the latter include things of
every day life as well as scientific concepts, as far as they are verifiable
by several independent experiments. The logical positivists who
emphatically claim to possess the only rigorous scientific philosophy,
as far as I understand, regard the constructs merely as conceptual
tools for surveying and ordering the crude sense data which alone
have the character of reality. These are minor variations of the
same theme. These variations appear to me unimportant, as two
essential points of reality are ignored. One such essential point is
that it is psychologically and physiologically wrong to regard the
crude sense impressions as the primary data; the other is that not
every concept from the domain of scientific constructs has the
character of a real thing, but only those which are invariant in
regard to the transformations involved.

With regard to the first point, we have to remember that every
human being has already acquired the ability to distinguish and
recognize objects in his first childhood. As a result, the world of a
normal human being is not a kaleidoscopic sequence of sensations
but a comprehensible, continuously changing scene of events in
which definite things preserve their identity, in spite of their ever
changing aspects. This power of the mind to neglect the differences
of sense impressions and to be aware only of their invariant features
seems to me the most impressive fact of our mental structure.
Imagine you are walking with your dog beside you. He sees a rabbit

1 62 PHYSICAL REALITY

and follows it in a wild chase, and soon the dog will be a tiny spot
in your field of vision. But all the time you see your dog, not a
sequence of visual impressions of diminishing size. Modern psychology
has recognized this fundamental situation; 1 mean the 'Gestalt'
psychology of KOHLER, HORNBOSTEL, WERTHEIMER, to name only a
few German psychologists of this school whom I personally knew.
I should like to translate the word 'Gestalt' not as 'shape' or 'form'
but as 'invariant', and speak of 'invariants of perception' as the
elements of our mental world. The physiology and anatomy of the
nervous system, of which I know a little from the writings of Pro-
fessor E. D. ADRIAN and Professor J. Z. YOUNG, are in full agreement
with this result of psychological observation.

Each single nerve fibre, whether motor or sensor, and in the latter
case whether carrying tactile, visual, auditory or thermal messages,
transfers a set of regular pulsations which have not the slightest
similarity to the physical stimulus. The brain receives nothing but
sequences of such pulsations, each propagated by a different fibre
to a definite place in the cortex, and it has the amazing ability to
disentangle these code messages almost instantaneously. What it
does is the solution of an extremely difficult problem of algebra,
determining the invariant features in this welter of ever-changing
signals. These features thus determine not a blurred set of impressions
but recognizable things.

If we attempted to build a philosophy of science on the assump-
tion that our raw material is unordered sense impressions, we could
not even describe our manipulations and simple instruments.
Science must accept, as I said before, the concepts of ordinary life
and the expressions of ordinary language. It transcends these by
using magnifying devices, telescopes, microscopes, electro-magnetic
amplifiers, etc. Thus new situations are encountered where ordinary
experience breaks down, and we are at a loss how to interpret the
signals received. You will understand what I mean if you have
ever looked through a microscope in which a medical friend is
showing you some remarkable cells or microbes: you see nothing
but a tangle of vague lines and colours and have to take his word
for it that some oval yellow structure is the object of interest.
Exactly the same happens in all branches of physics where amplifica-
tion is used. We glimpse the unknown, and we are bewildered. For
we are then not children any more; we have lost the power of
unconsciously decoding the nerve messages we are receiving, and
have to use our conscious technique of thinking, mathematics and
all its tricks (we except a few men of rare genius like FARADAY,
who saw the inner connection of nature by intuition like a child).

PHYSICAL REALITY 163

Thus we apply analysis to construct what is permanent in the flux
of phenomena, the invariants. Invariants are the concepts of which
science speaks in the same way as ordinary language speaks of
'things', and which it provides with names as if they were ordinary
things.

Of course, they are not. If we call an electron a particle we know
very well that it is not exactly like a grain of sand or pollen. For
instance, it has under certain circumstances not a distinct individu-
ality: if you shoot an electron out of an atom by another electron,
you can never tell which of the two electrons flying away is which.
Still it has some properties in common with ordinary 'particles',
thus justifying its name. Such extensions of nomenclature are quite
common in life as in science, and are systematically developed in
mathematics. A number means originally an integer with which
you can count a discrete set of objects. But the word is also used for
fractions like f, radicals like y^2, transcendentals like TT, and
imaginary numbers like <\/~ i, although you cannot count with
them. The justification is that they have some formal properties in
common with integers, each type a little less, but enough to use a
familiar word for them. The same principle is applied in analytical
geometry, when we speak of the infinitely distant line in a plane, or
of a four-dimensional sphere, and so on ; and also in physics. We
speak of infra-red or ultra-violet light although we cannot see it,
and of suprasonic sound although we cannot hear it. We are so
accustomed to extrapolate into regions beyond our sense qualities
that we have quite forgotten that we are extending concepts beyond
their original domain of definition. The principle of doing this is
always the same. Consider the concept of waves. We regard waves
on a lake as real, though they are nothing material but only a
certain shape of the surface of the water. The justification is that
they can be characterized by certain invariant quantities, like
frequency and wavelength, or a spectrum of these. Now the same
holds for light waves; why then should we withhold the epithet
'real 5 , even if the waves represent in quantum theory only a distribu-
tion of probability ? The feature which suggests reality is always
some kind of invariance of a structure independent of the aspect, the
projection. This feature, however, is the same in ordinary life and
in science, and the continuity between the things of ordinary life and
the things of science, however remote, compels us to use the same
language. This is also the condition for preserving the unity of pure
and applied science.

IS CLASSICAL MECHANICS IN FACT
DETERMINISTIC ?

[First published in Physikalische Blatter, vol. 1 1 (9), 49-54, 1955.]

THE laws of classical mechanics, and through them the laws of
classical physics as a whole, are so constructed that, if the
variables in a closed system are given at some initial point of time,
they can be calculated for any other instant in principle, at
least; for it is in most cases beyond human ability to carry out the
mathematics involved. This deterministic idea has greatly attracted
many thinkers, and has become an essential part of scientific
philosophy. Modern physics, however, has been compelled to
abandon determinism, together with other time-honoured theories
of space, time and matter, under the pressure of new empirical
discoveries. Quantum mechanics, which has taken over the place
of Newtonian mechanics, allows only statistical statements con-
cerning the behaviour of mass particles. The great majority of
physicists have become reconciled to this state of affairs, for it
corresponds exactly to the empirical situation in atomic and nuclear
physics, where experiments are based fundamentally on the
counting of events. Among the theoreticians, however, there are
some who are not content, and they are indeed some of the great
ones to whom the quantum theory owes its origin and development.
So far as I know, PLANCK himself was always sceptical towards the
statistical interpretation of quantum mechanics. The same is true
of EINSTEIN; even today he continues to point out, by means of
ingenious examples, contradictions in this interpretation (and he is,
moreover, still more concerned with the resolution of the concept
of physical reality, which is closely involved with the problem of
determinism). SCHRODESTGER goes still further; he proposes to
abandon the concept of particles (electrons, nuclei, atoms, etc.)
and to construct the whole of physics upon the idea of waves, which
obey deterministic laws in accordance with wave mechanics.
DE BROGLIE (and others) take the opposite course; they reject
waves, and seek a re-interpretation of quantum mechanics, in which
everything is in principle determinate, and an uncertainty in
prediction arises only by the presence.of concealed and unobservable
parameters. None of these physicists denies that quantum mechanics
within the realm of its validity (i.e. apart from the theory of

164

IS CLASSICAL MECHANICS IN FACT DETERMINISTIC? 165

elementary particles) is in agreement with experiment and meets all
the demands of the experimenters. Their rejection is in every case
founded on the assertion that the usual interpretation of the quantum
formulae is obscure and philosophically unsatisfactory.

What now is this philosophy? I do not think it can be traced
back before GALILEO and NEWTON. There were, of course, predic-
tions before that in astronomy, of conjunctions and eclipses, but the
men of antiquity and the Middle Ages saw order and predeter-
mination only in the celestial spheres, whilst caprice and chaos
reigned on earth. The religious tenets of fate and predestination
relate not to the processes of Nature, but to Man, and are certainly
fundamentally different from the mechanical determinism which
we here consider. The latter is inconceivable without NEWTON'S
laws of motion and their astonishing success in the prediction of
celestial events; it was derived from these laws, and later, during
the eighteenth and nineteenth centuries, became a fundamental
creed in science as a whole. The remarkable thing here is that the
undoubted fact that Newtonian mechanics does not suffice to
account for the observations, particularly in atomic physics, is
inadequate to shake belief in this abstract theorem.

But is it certain that classical mechanics in fact permits prediction
in all circumstances?* My doubts of this increase when I
compare the time scales of astronomy and atomic physics. The
age of the universe is reckoned to be some io 9 years, i.e. orbital
periods of the Earth. The number of periods in the ground state
of the hydrogen atom, on the other hand, is of the order of io 16
per second. Thus, when time is measured in the units appropriate
for each case, the situation is exactly the opposite of the simple
conception: the stellar universe is short-lived, and the atomic
universe extremely long-lived. Is it not dangerous to draw, from
experience of the short-lived universe, conclusions which are to be
valid for the long-lived one also?

These doubts are intensified when one considers the kinetic
theory of gases. It is usually asserted in this theory that the result
is in principle determinate, and that the introduction of statistical
considerations is necessitated only by our ignorance of the exact
initial state of a large number of molecules. I have long thought the
first part of this assertion to be extremely suspect. Let us consider
the simple case of a moving spherical molecule, which rebounds
elastically from numerous other fixed molecules (a kind of three-
dimensional bagatelle). A very small change in the direction of the

* The question was raised already by R. v. MISES; s. p. 17, article "On the
Meaning of Physical Theories", p. 34.

1 66 IS CLASSICAL MECHANICS IN FACT DETERMINISTIC?

initial velocity will then result in large changes of the path in the
zigzag motion; for a small angular change brings about larger and
larger spatial deviations, and so it must finally happen that a
sphere which was formerly hit is now missed. If the initial deviation
in direction is reduced, the moment when the path is changed to
another is delayed, but it will occur eventually. If we require
determinacy for all times, the smallest deviation in the initial
direction must be avoided.* But has this any physical meaning?
I am convinced that it has not, and that systems of this kind are
in fact indeterminate. To justify this assertion, a clear compre-
hension of the idea of determination is needed.

First of all, we may distinguish between dynamical stability
and instability. A motion is said to be stable if a small change
A# , Az> in the initial state (where x denotes the set of all co-
ordinates and v that of all velocities) causes only a small change
A#, Az; in the final state (so that, for all times, A* < Mkx , Az;
< AfA& , where M is a constant of the order of unity) . Otherwise
the motion is said to be unstable. It is fairly certain that the
motion of the spheres in the bagatelle game discussed above is
unstable. (This will be true a fortiori for a gas consisting of many
moving elastic particles.) The question has been much argued as to
whether or not the motion of the planets is stable. I do not know
what is the result of modern research (theory of the three-body
and many-body problems) ; it is of no importance for our purposes.
The essential thing is that there are systems which serve as models
of physical processes, and which, firstly, remain within a finite
region of space and for which, secondly, all motions are dynamically
unstable. The gas model which consists of elastic spheres in a
container with elastic walls is probably such a system, but it is too
complex to be analysed rigorously. It is sufficient to consider the
following trivially simple case. A mass particle moves without
friction along a straight line (the #-axis) under no forces, and is
elastically reflected at the termini (x = o, x = I). The co-ordinate
x remains in the finite interval o < x < / for any initial state
(*o> o)> the velocity v remains constant, but the deviation A*
increases with time (A* = A# + zAz> ) and takes arbitrarily large
values at sufficiently remote times. Thus any motion is unstable.

The connection with the problem of determinism is now evident.
If we wish to retain the assertion that in this system the initial
state determines every other state, we are compelled to demand

* We are evidently dealing with a double limit: the number of collisions tends
to infinity, while the change in direction tends to zero; the result is undetermined
in the absence of further data.

IS CLASSICAL MECHANICS IN FACT DETERMINISTIC? 167

absolutely exact values of # , v Q9 and to prohibit any deviation
A# , Az> . We could then speak of 'weak' determinacy, as opposed
to the 'strong' case where all motions are dynamically stable, and
therefore predictions are actually possible. This, however, would
be a mere evasion. The true situation is this. After a critical time
t c Z/Az> has been reached, the uncertainty A* > Z, and the
mass point may be found anywhere in the interval o < x < /.
That is to say, the final position is undetermined. If, however,
A0 is reduced, the critical time t c is only delayed ; it remains finite
for any finite A0 , and becomes infinite only for A0 = o, i.e. for an
absolutely definite initial velocity.

The connection with the problem of the continuum is evident
here. An exhaustive discussion of this question would take us too
far afield, and the following brief remarks must suffice. State-
ments like C A quantity x has a completely definite value' (expressed
by a real number and represented by a point in the mathematical
continuum) seem to me to have no physical meaning. Modern
physics has achieved its greatest successes by applying a principle
of methodology, that concepts whose application requires distinc-
tions that cannot in principle be observed, are meaningless and
must be eliminated. The most striking examples are EINSTEIN'S
foundation of the special and general theories of relativity (of which
the first rejects the concept of absolute simultaneity, and the second
the distinction between gravity and acceleration as unobservable),
and HEISENBERG'S foundation of quantum mechanics (by eliminating
the unobservable orbital radii and frequencies from BOHR'S theory
of the atom). The problem of continuity calls for the application
of the same principle. A statement like x = TT cm. would have a
physical meaning only if one could distinguish between it and
x = 7T n cm. for every n, where n n is the approximation of n by the
first n decimals. This, however, is impossible; and even if we
suppose that the accuracy of measurement will be increased in
the future, n can always be chosen so large that no experimental
distinction is possible.

Of course, I do not intend to banish from physics the idea of a
real number. It is indispensable for the application of analysis.
What I mean is that a physical situation must be described by means
of real numbers in such a way that the natural uncertainty in all
observations is taken into account.

Fifty years ago, FELIX KLEIN called for a similar step to be taken
in geometry. Besides abstract, exact geometry, he desired to have
a practical geometry, in which a point is replaced by a small spot,
straight lines by narrow strips, etc. However, nothing much

1 68 IS CLASSICAL MECHANICS IN FACT DETERMINISTIC?

resulted from this. In the meantime, physics has independently
developed the necessary tool, namely physical statistics. The
statement 'x is equal to a real number' is replaced by c The proba-
bility that x lies in an interval x < x < x z is P(x l \ x \ # 2 ).' Here
x, x l3 # 2 , P can be regarded as real numbers, since this is analytically
convenient, whilst the exact measurability of quantities is not
involved; P represents only the approximate expectation when
cases are counted for which x is limited approximately by x 1 and
x z . In other words, the true physical variable is the probability
density P(x).

Quantum mechanics has realized that this is the only possible
description of physical situations. (However, by introducing
probability amplitudes, it goes far beyond this statistical view-
point.)

In classical mechanics, the statistical method is used only for
systems of very many individual particles. Our model shows that
it is obligatory to use it in every case, even that of a single particle
in the simplest conceivable conditions. This does not require any
new mathematical considerations; for the law whereby the prob-
ability density varies is given at once by LIOUVTLLE'S theorem in
mechanics.* I shall elsewhere discuss exhaustively the mathe-
matical details and the relation to quantum mechanics. Here I
shall briefly give some results.

If we first continue to use classical mechanics, we find that our
model is perhaps the simplest example of the so-called ergodic
theorem of statistical mechanics. It can be very easily shown that
an initial probability density, describing an almost definite state,
passes in time into what is called the microcanonical distribution.
This therefore occurs automatically, even for one particle, and has
nothing to do with the 'large number' of particles. Complex
systems with energy exchange need be taken into account only if
we wish to pass to the canonical distribution.

Now, the same model can also be treated by quantum mechanics.
An initial state with an uncertainty A# in the initial position is
then described by a wave packet; the uncertainty Az> in the
initial velocity cannot be supposed arbitrarily small, but is related
to A# by HEISENBERG'S uncertainty relation A# . Az> > A/am;
this holds for all times, the factors A* and Ar> varying with time. If
both AA: O and A0 can be made small (for large masses), the quantum
formulae are identical with the classical ones to a close approxi-
mation, and there is again a critical instant t c where the individual

* See Appendix. Also Proceedings of the Danish Academy, 30, No. 2, 1955.
(Festskript til Niels Bohr.)

IS CLASSICAL MECHANICS IN FACT DETERMINISTIC? 1 69

motion ceases and a state is entered which can be described only
statistically. This corresponds exactly to the usual description of a
motion, in quantum mechanics, by means of stationary waves,
which is thus the analogue of the classical microcanonical distri-
bution.

To summarize, we may say that it is not the introduction of the
indeterministic statistical description which places quantum mech-
anics apart from classical mechanics, but other features, above
all the concept of the probability density as the square of a
probability amplitude P = | ^ | 2 ; the phenomenon of probability
interference results from this, and therefore it is impossible to
apply without modification the idea of an 'object 3 to the mass
particles of physics : the concept of physical reality must be revised.
This, however, is beyond the scope of these elementary considera-
tions.

APPENDIX

LIOUVILLE'S theorem expresses the conservation of probability
density during the motion, and leads to the differential equation

3P_ 3HBP _ffldP , .

dt ~~ dx dp dp dsf ' (I)

where H is HAMILTON'S function. (The expression on the right is
the so-called Poisson bracket). The solution corresponding to an
initial state P(x, p, o) = F(x,p} is

P(*,P* = F Lf(*>P> 0. (*> A *)]> - (*)

where f(x,p, t) = constant, g(x, p, t) = constant are two integrals
of the canonical equations of motion, normalized so that

f(x,p, o) = x, g(x,p, o) = A . . (3)

The solution of the probability equation (i) and of the canonical
equations thus present entirely equivalent problems. Nevertheless,
the solution of (i) furnishes new and interesting results.

For the example given in the text we have H = P 2 l2m; thus (i)
becomes

f - (*=^>- (4>
Two normalized integrals are^/ = x vt, g = v, and so the solution

(2) is

P = F(x vt,v). (5)

M

170 IS CLASSICAL MECHANICS IN FACT DETERMINISTIC?

The boundary conditions amount to the requirement of periodicity
in #(with period 2/) and antisymmetry in x and v :

F(x + al,v)=F(x,*),F(-x,-v)=F(x,v). . (6)
This can be satisfied with an arbitrary function /(#, 0) by

F(x, ) = S [f( 2 kl + x, v) +f(ati - x, - v)]. . (7)

k= CO

If we here replace x by x vt according to (5), we obtain
P(x,v 9 t). If the position and velocity at the initial instant are
almost definite, /(#, v) must be taken as a function having a sharp
maximum at (* , P O ) and vanishingly small elsewhere. If / is a
Gaussian function in both x (width <T O ) and v (width T O ), the resultant
^-distribution

P(x, = f ^(*> *>> * - (8)

is again a sum of Gaussian functions in x with width

* 2 ), . . (9)

which varies as f when t is large.

This passage to the limit t -^ co can be simply described by draw-
ing a small circle round the point (x 0) ) in the (x, p) phase space
(or the #0-plane), and examining how this breaks up into two
ellipses of equal area with centres # v Q t 9 whose major axes
become more and more parallel to the #-axis and finally longer
than the interval /.

ASTRONOMICAL RECOLLECTIONS

[First published in Vistas in Astronomy, Vol. i, pp. 41-44, 1955, Pergamon Press,
London. This work is dedicated to Professor F. J. M. STRATTON for the occasion
of his seventieth birthday.]

T AM not an astronomer, nor have I done any work in physics
* applicable to astronomy. Yet I cannot resist the wish to be
included amongst those who offer their congratulations to Professor
STRATTON by an article in this volume. There was a time in my life
when I was very near to devoting myself to the celestial science;
but I failed. May I offer, as a substitute for a more serious contribu-
tion, the story of my wrangling with astronomy and some recollec-
tions of remarkable astronomers who were my teachers.

I have to begin with Professor FRANZ, the director of the observa-
tory of my home city, Breslau. My father, who died just before I
finished school, had left me the advice to attend lectures on various
subjects before choosing a definite study for a profession. In Germany
at that period this was possible because of the complete 'academic
freedom' at the university.

There was in most subjects no strict syllabus, no supervision of
attendance, no examinations except the final ones. Every student
could select the lectures he liked best; it was his own responsibility
to build up a body of knowledge sufficient for the final examinations
which were either for a professional certificate or for a doctor's
degree, or both. Thus I made up a rather mixed programme for
my first year, including physics, chemistry, zoology, general philo-
sophy and logic, mathematics and astronomy. At school I had never
been very good nor interested in mathematics, but at the university
the only lectures which I really enjoyed were the mathematical and
astronomical ones. The greatest disappointment were the philo-
sophical courses; there we heard a lot about the rules of rational
thinking, the paradoxes of space, time, substance, cause, the
structure of the universe, and infinity. Yet it seemed to me an
awful muddle. Now the same concepts appeared also in the mathe-
matical and astronomical lectures, but instead of being veiled in a
mist of paradox they were formulated in a clear way according to
the case. For that was the important discovery I then made: that
all the high-sounding words connected with the concept of infinity
mean nothing unless applied in a definite system of ideas to a
definite problem.

171

172 ASTRONOMICAL RECOLLECTIONS

Astronomy was attractive in another way. There the problems
of cosmology are related to the infinity of the physical universe.
But little about these great questions was mentioned in the elemen-
tary lectures of our Professor FRANZ. What we had to learn was
the careful handling of instruments, correct reading of scales,
elimination of errors of observation and precise numerical calcula-
tions all the paraphernalia of the measuring scientist. It was a
rigorous school of precision, and I enjoyed it. It gave one the
feeling of standing on solid ground. Yet actually this feeling was not
quite justified by facts. The Breslau observatory was not on solid
ground, but on the top of the high and steep roof of the lovely
university building, in a kind of roof pavilion, decorated with
fantastic baroque ornaments and statues of saints and angels. The
main instrument was a meridian circle, which a hundred years ago
had been used by the great BESSEL; although it was placed on a
solid pillar standing on the foundations and rising straight through
the whole building, it was not free from vibrations produced by
the gales blowing from the Polish steppes. The whole outfit of this
observatory was old-fashioned and more romantic than efficient.
There were several old telescopes from WALLENSTEIN'S time, like
those KEPLER may have used. We had no electric chronograph
but had to learn to observe the stars crossing the threads in the
field of vision by counting the beats of a big clock and estimating
the tenths of a second. It was a very good school of observation, and
it had the additional attraction of an old and romantic craft.

I remember many an icy winter's night spent there in the little
roof pavilion. We were only three students in astronomy, and we
took the observations alternately. When my turn was finished I
enjoyed looking down on the endless expanse of snow-covered,
gabled roofs of the ancient city, the silhouettes against the starry
sky of the massive towers of the churches around the market place
and of the Cathedral further away beyond the river. There on the
narrow balcony amongst the stucco saints and old-fashioned
telescopes, one felt like an adept of Dr. Faustus and would not
have wondered if Mephistopheles had appeared behind the next
pillar. However, it was only old Professor FRANZ who came up the
steps to look after his three students he had not had so many for
a long time and who carried with him the soberness of the exact
scientist, checking our results and criticizing our endeavours with
mild and friendly irony.

These, our results, I rather think were not very reliable; it was
not so much our fault as that of the exalted but exposed position
of the observatory. Professor FRANZ himself, therefore, abstained

ASTRONOMICAL RECOLLECTIONS 1 73

from doing research, which needed exact measurements, and
restricted himself to descriptive work, a thorough study of the
moon's surface which he knew better than the geography of our
own planet He made strenuous efforts, however, to obtain a
modern observatory but never succeeded. During my student time
there were great hopes. The firm Carl Zeiss, Jena, had sent a set of
modern instruments to the World's Fair at Chicago. After the end
of the show these were purchased by the Prussian State for its
university observatories. Breslau obtained an excellent meridian
instrument and a big parallactic telescope; yet no proper building
was granted, and the meridian circle was installed in a wooden
cabin on a narrow island of the Oder River, just opposite the
university building. This island was in fact an artificial dam between
the river and a lock through which many barges used to pass. The
time service for the province of Silesia, which had been practised
for scores of years with the help of the old BESSEL circle, was trans-
ferred to the new Zeiss instruments, but the results remained highly
unsatisfactory. Eventually we discovered a correlation between the
strange irregularities of the time observations with the changing
level of the water in the lock; the island suffered small displacements
through the water pressure. Professor FRANZ'S hopes of a more
efficient observatory had broken down again.

We youngsters took this disappointment rather as a funny inci-
dent. It did not diminish the fascination which astronomy exerted
on my mind. This fascination was, however, shattered by the
horrors of computation. FRANZ gave us a lecture on the determina-
tion of planetary orbits, connected with a practical course where
we had to learn the technique of computing, filling in endless
columns of seven decimal logarithms of trigonometric functions
according to traditional forms. I knew from school that I was
bad at numerical work, but I tried hard to improve. It was in
vain, there was always a mistake somewhere in my figures, and my
results differed from those of the class mates. I was teased by them,
but that made it worse. I do not think that I ever finished an orbit
or an ephemeris, and then I gave up not only this calculating
business but the whole idea of becoming an astronomer. If I had
known at that time that there was in existence another kind of
astronomy which did not consider the prediction of planetary
positions as the ultimate aim, but studied the physical structure of
the universe with all the powerful instruments and concepts of
modern physics, my decision might have been different. But I
came in contact with astrophysics only some years later, when it
was too late to change my plans.

174 ASTRONOMICAL RECOLLECTIONS

At that period German students used to move from one university
to another, from different motives. Sometimes they were attracted
by a celebrated professor or a well-equipped laboratory, in other
cases by the amenities and beauties of a city, by its museums,
concerts, theatres, or by winter sport, by carnival and gay life in
general. Thus I spent two summer semesters in Heidelberg and
Zurich, returning during the winter to the home university. The
observatory of Heidelberg was on the Konigstuhl, a considerable,
wooded hill, where the astronomers lived a secluded life remote
from the ordinary crowd. I had then definitely changed over to
physics, and not even the celebrated name of WOLF, the professor
who has discovered more planetoids than anybody else, deflected
me from my purpose.

The observatory in Zurich was more accessible, and the name of
the professor was WOLFER, which could be interpreted as a com-
parative to WOLF. But even that did not attract me.

The following summer I went to Gottingen for the rest of my
student time. There KARL SCHWARZSCHILD was director of the
famous observatory which had been for many years under the
great GAUSS. SGHWARZSCHILD was the youngest professor of the
university, about thirty years of age; a small man with dark hair
and a moustache, sparkling eyes and an unforgettable smile. I
joined his astrophysical seminar and was for the first time intro-
duced to the modern aspect of astronomy. We discussed the atmos-
phere of planets, and I had to give an account of the loss of gas
through diffusion against gravity into interstellar space. Thus I
was driven to a careful study of the kinetic theory of gases which
then, in 1904, was not a regular part of the syllabus in physics.
But this is not the only subject which I first learned through
SGHWARZSGHILD'S teaching. His was a versatile, all-embracing mind,
and astronomy proper only one field of many in which he was
interested. About this time he published deep investigations on
electro-dynamics, in particular on the variational principle from
which LORENTZ'S equations for the field of an electron and for its
motion could be derived. In the following year (1905) there ap-
peared the first of his great articles on the aberrations of optical
instruments; these are, in my opinion, classical investigations, unsur-
passed in clarity and rigour by later work. I have presented this
method in my book Optik (Springer, 1932), and it is again to be
the backbone of a modernized version which will appear soon as
an English book on optics (in collaboration with E. WOLF*).

* Pergamon Press, London. To be published.

ASTRONOMICAL RECOLLECTIONS 175

SCHWARZSGHILD applied his aberration formulae to the actual
construction of new types of optical systems; but I am not com-
petent to speak about this part of his activities. Nor can I discuss
his astronomical work, experimental or theoretical. Personally he
was a most charming man, always cheerful, amusing, slightly
sarcastic, but kind and helpful. He once saved me from an awkward
situation. I had intended to take geometry as one of my subjects in
the oral examinations for the doctor's degree, but was not attracted
by the lectures of FELIX KLEIN, the famous mathematician, and
attended somewhat irregularly. This fact did not escape KLEIN'S
observation and he showed me his displeasure. A disaster at the
orals, only six months ahead, seemed to be impending. But
SCHWARZSGHILD said that half a year was ample time to learn the
whole of astronomy. He gave me some books to read and tutored
me a little, in exchange for my training him in tennis. When the
examination came his first question was: What do you do when
you see a falling star?' Whereupon I answered at once: C I make a
wish' according to an old German superstition that such a wish
is always fulfilled. He remained quite serious and continued:
'Yes, and what do you do then?' Whereupon I gave the expected
answer: e l would look at my watch, remember the time, constella-
tion of appearance, direction of motion, range, etc., go home and
work out a crude orbit'. Which led to celestial mechanics and to a
satisfactory pass. SCHWARZSGHILD differed from the ordinary type
of the dignified, bearded German scholar of that time; not only in
appearance, but also in his mental structure, which was thoroughly
modern, cheerful, active, open to all problems of the day. Still he
had his hours of professorial absent-mindedness. There was a
'Stammtisch', a certain table in a restaurant where a group of
young professors and lecturers used to meet for lunch. SCHWARZ-
SGHILD was one of them until his marriage. A few weeks after the
wedding he was again at his accustomed place at the lunch table
and plunged in his usual way into a lively discussion about some
scientific problem, until one of the men asked him: 'Now, SCHWARZ-
SCHILD, how do you like married life?' He blushed, jumped up, said:
* Married life oh, I have quite forgotten ', got his hat and ran
away. But I think this kind of behaviour was not typical of him.
He always knew what he was doing. His life was short, his achieve-
ments amazing, his success great his end tragic. When the great
war of 1914-18 broke out he was employed as a mathematical
expert in ballistics and attached to the staff of one of the armies
on the Eastern front. There, in Russia, he contracted some rare
infectious disease. It was said that he refused to be sent home, until

176 ASTRONOMICAL RECOLLECTIONS

it was too late. On his way home, he visited me in my military
office in Berlin; he was still cheerful, but he looked terribly ill.
Soon after he died. Now his son, Martin, keeps up the astronomical
tradition, thus founding another one of those hereditary lines of
astronomers, the HERSGHELS, the STRUVES, and so on,

I have met many other distinguished astronomers and been
intimate with some of them; but as most of them are still wandering
on this globe, I had better refrain from telling stories about them.

May I conclude by wishing Professor STRATTON many happy
returns and by adding the request that he too may present us with
some recollections of astronomical personalities out of his long
experience.

STATISTICAL INTERPRETATION OF
QUANTUM MECHANICS

[First published in Science, Vol. 122, No. 3172, pp. 675-679 (1955). This
article is the English translation of the lecture Professor BORN gave in German
when he was awarded the Nobel Prize for Physics in 1954, a prize which he
shared with W. BOTHE.]

'""pHE published work for which the honour of the Nobel prize
-*- for the year 1954 has been accorded to me does not contain
the discovery of a new phenomenon of nature but, rather, the
foundations of a new way of thinking about the phenomena of
nature. This way of thinking has permeated experimental and
theoretical physics to such an extent that it seems scarcely possible
to say anything more about it that has not often been said already.
Yet there are some special aspects that I should like to discuss.

The first point is this : The work of the Gottingen school, of which
I was at that time the director, during the years 1926 and 1927,
contributed to the solution of an intellectual crisis into which our
science had fallen through PLANCK'S discovery of the quantum of
action in the year 1900. To-day physics is in a similar crisis I do
not refer to its implication in politics and economics consequent on
the mastery of a new and terrible force of nature, but I am thinking
of the logical and epistemological problems posed by nuclear physics.
Perhaps it is a good thing to remind oneself at such a time of what
happened earlier in a similar situation, especially since these events
are not without a certain element of drama. In the second place,
when I say that physicists had accepted the way of thinking deve-
loped by us at that time, I am not quite correct. There are a few
most noteworthy exceptions namely, among those very workers
who have contributed most to the building up of quantum theory.
PLANCK himself belonged to the sceptics until his death. EINSTEIN,
DE BROGLIE, and SCHRODINGER have not ceased to emphasize the
unsatisfactory features of quantum mechanics, and to demand a
return to the concepts of classical, Newtonian physics, and to
propose ways in which this could be done without contradicting
experimental facts. One cannot leave such weighty views unheard.
NIELS BOHR has gone to much trouble to refute the objections. I
have myself pondered on them and believe I can contribute some-
thing to the clarification of the situation. We are concerned with
the borderland between physics and philosophy, and so my physical

178 STATISTICAL INTERPRETATION OF QUANTUM MECHANICS

lecture will be partly historically and partly philosophically coloured,
for which I ask indulgence.

First of all, let me relate how quantum mechanics and its
statistical interpretation arose. At the beginning of the 1 920*3
every physicist, I imagine, was convinced that PLANCK'S hypothesis
was correct, according to which the energy in oscillations of definite
frequency v (for example, in light waves) occurs in finite quanta
of size hv. Innumerable experiments could be explained in this
manner and always gave the same value of PLANCK'S constant h.
Furthermore, EINSTEIN'S assertion that light quanta carry momen-
tum hv\c (where c is the velocity of light) was well supported by
experiment. This meant a new lease of life for the corpuscular
theory of light for a certain complex of phenomena. For other
processes, the wave theory was appropriate. Physicists accustomed
themselves to this duality and learned to handle it to a certain
extent.

In 1913 NIELS BOHR had solved the riddle of line spectra by using
quantum theory and at the same time had explained, in their
main features, the wonderful stability of atoms, the structure of
their electronic shells, and the periodic system of the elements.
For the sequel the most important assumption of his teaching was
this: an atomic system cannot exist in all mechanically possible
states, which form a continuum, but in a series of discrete 'stationary'
states; in a transition from one to another the difference in energy
E m E n is emitted or absorbed as a light quantum hv mn (according
as E m is greater or less than JE n ). This is an interpretation, in
terms of energy, of the fundamental law of spectroscop'y discovered
some years previously by W. Rrrz. The situation can be pictured
by writing the energy levels of the stationary states twice over,
horizontally and vertically; a rectangular array results

II 12 13

21 22 23

in which positions on the diagonal correspond to the states and off-
diagonal positions correspond to the transitions.

BOHR was fully aware that the law thus formulated is in conflict
with mechanics and that, therefore, even the use of the concept of
energy in this context is problematical. He based this bold fusion
of the old with the new on his principle of correspondence. This

STATISTICAL INTERPRETATION OF QUANTUM MECHANICS 179

consists in the obvious requirement that ordinary classical mech-
anics must hold to a high degree of approximation in the limit,
when the numbers attached to the stationary states, the quantum
numbers, are very large that is, far to the right and low down in
the foregoing array so that the energy changes relatively little
from place to place that is, practically continuously.

Theoretical physics lived on this idea for the next 10 years.
The problem was that a harmonic oscillator possesses not only
frequency but intensity as well. For each transition in the scheme
there must be a corresponding intensity. How is the latter to be
found by considerations of correspondence ? It was a question of
guessing the unknown from a knowledge of a limiting case. Con-
siderable success was achieved by BOHR himself, by KRAMERS, by
SOMMERFELD, by EPSTEIN, and by many others. But the decisive
step was again taken by EINSTEIN, who, by a new derivation of
PLANCK'S radiation formula, made it evident that the classical con-
cept of intensity of emission must be replaced by the statistical idea
of transition probability. To each position in our scheme there
belongs, besides the frequency v mn = (E m W )/A, a certain
probability for the transition accompanied by emission or absorption
of radiation.

In Gottingen we also took part in the attempts to distill the
unknown mechanics of the atom out of the experimental results.
The logical difficulty became ever more acute. Investigations on
scattering and dispersion of light showed that EINSTEIN'S conception
of transition probability as a measure of the strength of an oscillation
was not adequate, and the idea of an oscillation amplitude asso-
ciated with each transition could not be dispensed with. In this
connection work by LADENBURG [i], KRAMERS [2], HEISENBERG [3],
JORDAN and I [4] may be mentioned. The art of guessing correct
formulas, which depart from the classical formulas but pass over
into them in the sense of the correspondence principle, was brought
to considerable perfection. A paper of mine, which introduced in
its title the expression 'quantum mechanics', probably for the first
time, contains a very involved formula still valid at the present
time for the mutual disturbance of atomic systems.

This period was brought to a sudden end by HEISENBERG [5],
who was my assistant at that time. He cut the Gordian knot by a
philosophical principle and replaced guesswork by a mathematical
rule. The principle asserts that concepts and pictures that do not
correspond to physically observable facts should not be used in
theoretical description. When EINSTEIN, in setting up his theory of
relativity, eliminated the concepts of the absolute velocity of a

l8o STATISTICAL INTERPRETATION OF QUANTUM MECHANICS

body and of the absolute simultaneity of two events at different
places, he was making use of the same principle. HEISENBERG
banished the picture of electron orbits with definite radii and periods
of rotation, because these quantities are not observable; he
demanded that the theory should be built up by means of quadratic
arrays of the kind suggested in a preceding paragraph. Instead of
describing the motion by giving a co-ordinate as a function of time
x(i), one ought to determine an array of transition probabilities
x mn . To me the decisive part in his work is the requirement that one
must find a rule whereby from a given array

the array for the square,
(* 2 )u

may be found (or, in general, the multiplication law of such arrays).

By consideration of known examples discovered by guesswork he
found this rule and applied it with success to simple examples such
as the harmonic and anharmonic oscillator. This was in the summer
of 1925. HEISENBERG, suffering from a severe attack of hay fever,
took leave of absence for a course of treatment at the seaside and
handed over his paper to me for publication, if I thought I could
do anything about it.

The significance of the idea was immediately clear to me, and I
sent the manuscript to the %eitschrift fur Physik. HEISENBERG'S rule
of multiplication left me no peace, and after a week of intensive
thought and trial, I suddenly remembered an algebraic theory that
I had learned from my teacher, ROSANES, in Breslau. Such quadratic
arrays are quite familiar to mathematicians and are called matrices,
in association with a definite rule of multiplication. I applied this
rule to HEISENBERG'S quantum condition and found that it agreed
for the diagonal elements. It was easy to guess what the remaining
elements must be, namely, null; and immediately there stood before
me the strange formula

pq qp A/2?n.

STATISTICAL INTERPRETATION OF QUANTUM MECHANICS l8l

This meant that co-ordinates q and momenta p are not to be repre-
sented by the values of numbers but by symbols whose product
depends on the order of multiplication which do not 'commute 9 ,
as we say.

My excitement over this result was like that of the mariner who,
after long voyaging, sees the desired land from afar, and my only
regret was that HEISENBERG was not with me. I was convinced
from the first that we had stumbled on the truth. Yet again a large
part was only guesswork, in particular the vanishing of the non-
diagonal elements in the foregoing expression. For this problem I
secured the collaboration of my pupil PASCUAL JORDAN, and in a
few days we succeeded in showing that I had guessed correctly.
The joint paper by JORDAN and myself [6] contains the most
important principles of quantum mechanics, including its extension
to electrodynamics.

There followed a hectic period of collaboration among the three
of us, rendered difficult by HEISENBERG'S absence. There was a
lively interchange of letters, my contribution to which unfortunately
went amiss in the political disorders. The result was a three-man
paper [7], which brought the formal side of the investigation to a
certain degree of completeness. Before this paper appeared, the
first dramatic surprise occurred: PAUL DIRAC'S paper [8] on the
same subject. The stimulus received through a lecture by HEISEN-
BERG in Cambridge led him to results similar to ours in Gottingen,
with the difference that he did not have recourse to the known matrix
theory of the mathematicians but discovered for himself and elab-
orated the doctrine of such non-commuting symbols.

The first non-trivial and physically important application of
quantum mechanics was made soon afterwards by W. PAULI [9],
who calculated the stationary energy values of the hydrogen atom
by the matrix method and found complete agreement with BOHR'S
formulas. From this moment there was no longer any doubt about
the correctness of the theory.

What the real significance of this formalism might be was, how-
ever, by no means clear. Mathematics, as often happens, was
wiser than interpretative thought. While we were still discussing
the point, there occurred the second dramatic surprise: the appear-
ance of SCHRODINGER'S celebrated papers [10]. He followed quite
a different line of thought, which derived from Louis DE BROGLIE
[i i]. The latter had a few years previously made the bold assertion,
supported by brilliant theoretical considerations, that wave-
corpuscle dualism, familiar to physicists in the case of light, must
also be exhibited by electrons; to each freely movable electron

1 82 STATISTICAL INTERPRETATION OF QUANTUM MECHANICS

there belongs, according to these ideas, a plane wave of perfectly
definite wavelength, determined by PLANCK'S constant and the mass.
This exciting essay by DE BROGLIE was well known to us in Got-
tingen.

One day in 1925 I received a letter from C. J. DAVISSON con-
taining singular results on the reflection of electrons from metallic
surfaces. My colleague on the experimental side, JAMES FRANCK,
and I at once conjectured that these curves of DAVISSON'S were
crystal-lattice spectra of DE BROGLIE'S electron waves, and we
arranged for one of our pupils, W. ELSASSER [12], to investigate
the matter. His result provided the first quantitative proof of DE
BROGLIE'S idea, a proof independently given later by DAVISSON and
GERMER [13] and by G. P. THOMSON [14], by systematic experi-
ments.

But this familiarity with DE BROGLIE'S line of thought did not
lead on further toward an application to the electronic structure of
atoms. This was reserved for SCHRODINGER. He extended DE
BROGLIE'S wave equation, which applied to free motion, to the
case in which forces act and gave an exact formulation of the
additional conditions, already hinted at by DE BROGLIE, to which the
wave function ijr must be subjected namely, that it should be
single-valued and finite in space and time and he succeeded in
deriving the stationary states of the hydrogen atom as mono-
chromatic solutions of his wave equation not extending to infinity.
For a short while, at the beginning of 1926, it looked as if suddenly
there were two self-contained but entirely distinct systems of ex-
planation in the field matrix mechanics and wave mechanics.
But SCHRODINGER himself soon demonstrated their complete
equivalence.

Wave mechanics enjoyed much greater popularity than the
Gottingen or Cambridge version of quantum mechanics. Wave
mechanics operates with a wave function ^, which at least in
the case of one particle can be pictured in space, and it employs
the mathematical methods of partial differential equations familiar
to every physicist. SCHRODINGER also believed that his wave theory
made possible a return to deterministic classical physics; he pro-
posed (and has emphatically renewed this suggestion quite recently,
[i 5]) to abandon the particle picture entirely and to speak of electrons
not as particles but as a continuous ^density distribution |^| 2 ,
or electric density e\ i/r | 2 .

To us in Gottingen this interpretation appeared unacceptable in
the face of the experimental facts. At that time it was already possible
to count particles by means of scintillations or with the Geiger

STATISTICAL INTERPRETATION OF QUANTUM MECHANICS 183

counter and to photograph their tracks with the help of the Wilson
cloud chamber.

It appeared to me that it was not possible to arrive at a clear
interpretation of the ^-function by considering bound electrons. I
had therefore been at pains, as early as the end of 1925, to extend
the matrix method, which obviously covered only oscillatory pro-
cesses, in such a way as to be applicable to aperiodic processes. I
was at that time the guest of the Massachusetts Institute of Tech-
nology in the U.S.A., and there I found in NORBERT WIENER a
distinguished collaborator. In our joint paper [16] we replaced the
matrix by the general concept of an operator and, in this way,
made possible the description of aperiodic processes. Yet we missed
the true approach, which was reserved for SGHRODINGER; and I
immediately took up his method, since it promised to lead to an
interpretation of the ^-function. Once more an idea of EINSTEIN'S
gave the lead. He had sought to make the duality of particles (light
quanta or photons) and waves comprehensible by interpreting the
square of the optical wave amplitudes as probability density for the
occurrence of photons. This idea could at once be extended to the
^-function: \i/r\ 2 must represent the probability density for
electrons (or other particles). To assert this was easy; but how was
it to be proved?

For this purpose atomic scattering processes suggested themselves.
A shower of electrons coming from an infinite distance, represented
by an incident wave of known intensity (that is, |^| 2 ) impinge
on an obstacle, say a heavy atom. In the same way that the water
wave caused by a steamer excites secondary circular waves in
striking a pile, the incident electron wave is partly transformed by
the atom into a secondary spherical wave, whose amplitude of
oscillation ft is different in different directions. The square of the
amplitude of this wave at a great distance from the scattering centre
then determines the relative probability of scattering in its depend-
ence on direction. If, in addition, the scattering atom is itself
capable of existing in different stationary states, one also obtains
quite automatically from SCHRODINGER'S wave equation the pro-
babilities of excitation of these states, the electron being scattered
with loss of energy, or inelastically, as it is termed. In this way it
was possible to give the assumptions of BOHR'S theory, first verified
experimentally by FRANCK and HERTZ, a theoretical basis [17].
Soon WENTZEL [18] succeeded in deriving RUTHERFORD'S celebrated
formula for the scattering of a-particles from my theory.

But the factor that contributed more than these successes to the
speedy acceptance of the statistical interpretation of the ^-function

184 STATISTICAL INTERPRETATION OF QUANTUM MECHANICS

was a paper by HEISENBERG [19] that contained his celebrated
uncertainty relationship, through which the revolutionary character
of the new conception was first made clear. It appeared that it was
necessary to abandon not only classical physics but also the naive
conception of reality that thought of the particles of atomic physics
as if they were exceedingly small grains of sand. A grain of sand
has at each instant a definite position and velocity. For an electron
this is not the case; if one determines the position with increasing
accuracy, the possibility of determining the velocity becomes less,
and vice versa. I shall return to these questions in a more general
connection, but before doing so would like to say a few words about
the theory of collisions.

The mathematical techniques of approximation I used were
somewhat primitive and were soon improved. Out of the literature,
which has grown to unmanageable proportions, I can name only
a few of the earliest authors, to whom the theory is indebted for
considerable progress: HOLTSMARK in Norway, FAXEN in Sweden,
BETHE in Germany, MOTT and MASSEY in Great Britain.

To-day collision theory is a special science, with its own volumin-
ous text-books, and has grown completely over my head. Of course,
in the last resort all the modern branches of physics, quantum
electrodynamics, the theory of mesons, nuclei, cosmic rays, ele-
mentary particles and their transformations, all belong to this
range of ideas, to a discussion of which no bounds could be set.

I should also like to state that during the years 1926 and 1927
I tried another way of justifying the statistical conception of quan-
tum mechanics, partly in collaboration with the Russian physicist
FOCK [20]. In the afore-mentioned three-man paper there is a
chapter in which the SCHRODINGER function is really anticipated;
only it is not thought of as a function ft of space, but as function ft n
of the discrete index n = i, 2, ... which enumerates the
stationary states. If the system under consideration is subject to a
force that is variable in time, i/r n also becomes time-dependent, and
| ft n (t) \ 2 denotes the probability for the existence of that state
n at time t.

Starting from an initial distribution in which only one state is
present, we obtain in this manner transition probabilities, and we
can investigate their properties. In particular, what interested me
most at the time was what happens in the adiabatic limiting case,
that is, in the case of very slowly variable external action; it was
possible to show that, as might have been expected, the probability
of transitions became ever smaller. The theory of transition
probabilities was developed independently by DIRAG and made to

STATISTICAL INTERPRETATION OF QUANTUM MECHANICS 185

yield results. It may be said that the whole of atomic and nuclear
physics works with this system of concepts, especially in the extremely
elegant form given to them by DIRAC [21] ; almost all experiments
lead to statements about relative probabilities of events, even if they
appear concealed under the name cross section or the like.

How then does it come about that great discoverers such as
EINSTEIN, SCHRODINGER, and DE BROGUE are not satisfied with the
situation? As a matter of fact, all these objections are directed not
against the correctness of the formulas but against their inter-
pretation. Two closely interwoven points of view must be distin-
guished : the question of determinism and the question of reality.

Newtonian mechanics is deterministic in the following sense. If
the initial state (positions and velocities of all particles) of a system
is accurately given, the state at any other tune (earlier or later)
may be calculated from the laws of mechanics. All the other
branches of classical physics have been built up in accordance with
this pattern. Mechanical determinism gradually became an article
of faith the universe as a machine, an automaton. As far as I
can see, this idea has no precursors in ancient or mediaeval phil-
osophy; it is a product of the immense success of Newtonian
mechanics, especially in astronomy. In the nineteenth century it
became a fundamental philosophic principle for the whole of exact
science. I asked myself whether this was really justified. Can we
really make absolute predictions for all time on the basis of the
classical equations of motion? It is easily seen, by simple examples,
that this is the case only if we assume the possibility of absolutely
accurate measurement (of the position, velocity, or other quantities).
Let us consider a particle moving without friction on a straight line
between two end-points (walls) at which it suffers perfectly elastic
recoil. The particle moves backward and forward with constant
speed equal to its initial speed z> , and one can say exactly where it
will be at a stated time provided that V Q is accurately known.

But if we allow a small inaccuracy Az> , the inaccuracy of the
prediction of position at time t is fAz; ; that is, it increases with t.
If we wait long enough, until time t c = Z/ Az> , where c is the dis-
tance between the elastic walls, the inaccuracy A* will have become
equal to the whole interval L Thus it is possible to say absolutely
nothing about the position at a time later than t c . Determinism
becomes complete indeterminism if one admits even the smallest
inaccuracy in the velocity datum. Is there any sense I mean
physical, not metaphysical, sense in which one can speak of abso-
lute data? Is it justifiable to say that the co-ordinate x is TT cm,
where TT = 3-1415 .- is the familiar transcendental number

N

1 86 STATISTICAL INTERPRETATION OF QUANTUM MECHANICS

that determines the ratio of the circumference of a circle to its
diameter? As an instrument of mathematics, the concept of a
real number represented by a nonterminating decimal is extremely
important and fruitful. As a measure of a physical quantity, the
concept is nonsensical. If the decimal for TT is interrupted at the
soth or 25th place, two numbers are obtained which cannot be
distinguished by any measurement from each other and from the
true value. According to the heuristic principle employed by
EINSTEIN in the theory of relativity and by HEISENBERG in quantum
theory, concepts that correspond to no conceivable observation
ought to be eliminated from physics. This is possible without
difficulty in the present case also; we have only to replace state-
ments like x = TT cm. by: the probability of the distribution of
values of # has a sharp maximum at x == TT cm.; and (if we wish to
be more accurate) we can add : of such and such a breadth. In
short, ordinary mechanics must be formulated statistically. I have
occupied myself with this formulation a little recently and have
seen that it is possible without difficulty. This is not the place to go
into the matter more closely. I only wish to emphasize the point
that the determinism of classical physics turns out to be a false
appearance, produced by ascribing too much weight to mathe-
maticological conceptual structures. It is an idol, not an ideal, in
the investigation of nature and, therefore, cannot be used as an
objection to the essentially indeterministic, statistical interpretation
of quantum mechanics.

Much more difficult is the objection concerned with reality. The
concept of a particle, for example, a grain of sand, contains impli-
citly the notion that it is at a definite position and has a definite
motion. But according to quantum mechanics it is impossible to
determine simultaneously with arbitrary accuracy position and
motion (more correctly momentum, that is, mass times velocity).
Thus two questions arise. First, what is there to prevent us from
measuring both quantities with arbitrary accuracy by refined
experiments, in spite of the theoretical assertion? Second, if it
should really turn out that this is not feasible, are we still justified
in applying to the electron the concept of particle and the ideas
associated with it?

With regard to the first question, it is clear that if the theory is
correct and we have sufficient grounds for believing this the
obstacle to simultaneous measurability of position and motion (and
of other similar pairs of so-called 'conjugate' quantities) must lie
in the laws of quantum mechanics itself. This is indeed the case,
but it is not at all obvious. NIELS BOHR himself has devoted much

STATISTICAL INTERPRETATION OF QUANTUM MECHANICS 187

labour and ingenuity to developing a theory of measurements to
clear up this situation and to meet the most subtle considerations
of EINSTEIN, who repeatedly tried to think out measuring devices
by means of which position and motion could be measured simul-
taneously and exactly. The conclusion is as follows. In order to
measure space co-ordinates and instants of time rigid measuring
rods and clocks are required. On the other hand to measure
momenta and energies arrangements with movable parts are
needed to take up and indicate the impact of the object to be
measured. If we take into consideration the fact that quantum
mechanics is appropriate for dealing with the interaction of object
and apparatus, we see that no arrangement is possible that satisfies
both conditions at the same time. There exist, therefore, mutually
exclusive but complementary experiments, which only in combina-
tion with each other disclose all that can be learned about an
object. This idea of complementarity in physics is generally regarded
as the key to the intuitive understanding of quantum processes.
BOHR has transferred the idea in an ingenious manner to completely
different fields for example, to the relationship between conscious-
ness and brain, to the problem of free will, and to other fundamental
problems of philosophy.

Now to come to the final point can we still call something with
which the concepts of position and motion cannot be associated in
the usual way a thing, a particle? And if not, what is the reality that
our theory has been invented to describe?

The answer to this question is no longer physics, but philosophy,
and to deal with it completely would overstep the bounds of this
lecture. I have expounded my views on it fully elsewhere [23].
Here I will only say that I am emphatically for the retention of the
particle idea. Naturally it is necessary to redefine what is meant.
For this purpose well-developed concepts are available, which are
familiar in mathematics under the name of invariants with respect
to transformations. Every object that we perceive appears in
innumerable aspects. The concept of the object is the invariant
of all these aspects. From this point of view, the present universally
used conceptual system, in which particles and waves occur at the
same time, can be completely justified.

The most recent research on nuclei and elementary particles has,
however, led us to limits beyond which this conceptual system in its
turn does not appear to suffice. The lesson to be learned from the
story I have told of the origin of quantum mechanics is that, pre-
sumably, a refinement of mathematical methods will not suffice to
produce a satisfactory theory, but that somewhere in our doctrine

1 88 STATISTICAL INTERPRETAHONJDF QUANTUM MECHANICS

there lurks a concept not justified by any experience, which will
have to be eliminated in order to clear the way.

REFERENCES

1. R. LADENBURG, . Physik 4, 451 (1921); R. LADENBURG and F. REICHE,

Naturwiss. 11, 584 (1923).

2. H. A. KRAMERS, Nature 113, 673 (1924).

3. and W. HEISENBERG, . Physik 31, 631 (1925).

4. M. BORN, ibid. 26, 379 (1924) ; M. BORN and P. JORDAN, ibid. 33, 479 (1925).

5. W. HEISENBERG, ibid. 33, 879 (1925).

6. M. BORN and P. JORDAN, ibid. 34, 358 (1925).

7. M. BORN, W. HEISENBERG, P. JORDAN, ibid. 35, 557 (1926).

8. P. A. M. DIRAC, Proc. Roy. Soc. (London] AHM), 643 (1925).

9. W. PAUIJ, Z- Physik 36, 336 (1926).

10. E. ScHRdDiNGER, Ann. Physik (4), 79, 361, 489, 734 (1926); 80, 437 (1926);

81, 109 (1926).
n. Louis DE BROGLIE, Theses, Paris, 1924; Ann. Physik (10), 3, 22 (1925).

12. W. ELSASSER, Naturwiss. 13, 711 (1925).

13. C. J. DAVISSON and L. H. GERMER, Phys. Rev. 30, 707 (1927).

14. G. P. THOMSON and A. RJEID, Nature 119, 890 (1927); G. P. THOMSON, Proc.

Roy Soc. (London) Aii7, 600 (1928).

15. E. SCHRODINGER, Brit. J. Phil. Sci. 3, 109, 233 (1952).

1 6. M. BORN and N. WIENER, . Physik 36, 174 (1926).

17. M. BORN, ibid. 37, 863 (1926); 38, 803 (1926); Gott. Nachr. Math.-Physik

Ki, i, 146 (1926).

1 8. G. WENTZEL, . Physik 40, 590 (1926).

19. W. HEISENBERG, ibid. 43, 172 (1927).

20. M. BORN, ibid. 40, 167 (1926); M. BORN and V. FOGK, ibid., 51, 165 (1928).

21. P. A. M. DIRAG, Proc. Roy. Soc. (London) Aiog, 642 (1925); no, 561 (1926);

in, 281 (1926); 112, 674 (1926).

22. NIELS BOHR, Naturwiss. 16, 245 (1928); 17, 483 (1929); ai, 13 (1933);

'Causality and complementarity', Die Erkemtnis 6, 293 (1936).

23. M. BORN, Phil. Quart. 3, 134 (1953); Physik. Bl. 10, 49 (1954).

PHYSICS AND RELATIVITY

[A lecture given at the International Relativity Conference in Berne, Switzerland,
on 1 6th July, 1955.]

T HAVE been honoured by being asked to give the address on
-Physics and Relativity in place of NIELS BOHR who was prevented
from coming to Berne.

I do not know what BOHR had in mind when he chose the title.
I cannot remember that I have ever discussed relativity with him;
there was in fact nothing to discuss as we agreed on all essential
points. The title Physics and Relativity may be interpreted in
different ways: it may mean either a review of the empirical facts
on which relativity was built, or it may mean a survey of the
consequences of relativity for the whole of physics. Now such a
survey was just the purpose of this conference, and it would be
presumptuous and quite beyond my power to summarize all the
reports and investigations. I propose instead to give you an impres-
sion of the situation of physics 50 years ago when EINSTEIN'S first
papers appeared, to analyse the contents of these papers in com-
parison with the work of his predecessors and to describe the impact
of them on the world of physics. For most of you this is history.
Relativity was an established theory when you began to study.
There are very few left who like me can remember those distant
days. For my contemporaries EINSTEIN'S theory was new and
revolutionary, an effort was needed to assimilate it. Not everybody
was able or willing to do so. Thus the period after EINSTEIN'S
discovery was full of controversy, sometimes of bitter strife. I
shall try to revive these exciting days when the foundation of
modern physics was laid, by telling the story as it appeared to me.

When I began to study in the year 1901 MAXWELL'S theory was
accepted everywhere but not taught everywhere. A lecture by
CLEMENS SCHAEFER which I attended at Breslau University was
the first of its kind there and appeared to us to be very difficult.
When I came to Gottingen in 1904 I attended a lecture on optics
by WOLDEMAR VOIGT, which was based on MAXWELL'S theory;
but that was a new venture, the transition from the elastic ether
theory was only a few years old. The main representative of the
modern spirit in theoretical physics at Gottingen was at that time
MAX ABRAHAM, whose well-known book, then called Abraham-
Foppl, now Abraham-Becker ', was our main source of information.

189

1 90 PHYSICS AND RELATIVITY

All this is to indicate the scientific atmosphere in which we grew up.
NEWTON'S mechanics still dominated the field completely, in spite
of the revolutionary discoveries made during the preceding decade.
X-rays, radio-activity, the electron, the radiation formula and the
quantum of energy, etc. The student was still taught and I
think not only in Germany, but everywhere that the aim of
physics was to reduce all phenomena to the motion of particles
according to NEWTON'S laws, and to doubt these laws was heresy
never attempted.

My first encounter with the difficulties of this orthodox creed
happened in 1905, the year which we celebrate to-day, in a seminar
on the theory of electrons, held not by a physicist but by a mathe-
matician, HERMANN MINKOWSKI. My memory of these long bygone
days is of course blurred, but I am sure that in this seminar we
discussed what was known at this period about the electrodynamics
and optics of moving systems. We studied papers by HERTZ,
FITZGERALD, LARMOR, LORENTZ, POINCARE, and others but also
got an inkling of MINKOWSKI'S own ideas which were published only
two years later.

I have now to say some words about the work of these predecessors
of EINSTEIN, mainly of LORENTZ and POINCARE. But I confess that
I have not read again all their innumerable papers and books.
When I retired from my chair at Edinburgh I settled at a quiet
place where no scientific library is available, and I got rid of most
of my own books. Therefore I rely a good deal on my own memory,
assisted by a few books which I shall quote.

H. A. LORENTZ' important papers of 1892 and 1895 on the electro-
dynamics of moving bodies contain much of the formalism of
relativity. However, his fundamental assumptions were quite un-
relativistic. He assumed an ether absolutely at rest, a kind of
materialization of NEWTON'S absolute space, and he also took
NEWTON'S absolute time for granted. When he discovered that his
field equations for empty space were invariant for certain linear
transformations, by which the co-ordinates x, j>, z and the time t
were simultaneously transformed into new parameters #', j/, z', t',
he called them *local co-ordinates' and c local time'. These trans-
formations, for which POINCAR later introduced the term Lorentz
transformations, were in fact older; already in 1887 W. VOIGT
had observed that the wave equation of the elastic theory of light
was invariant with respect to this type of transformations. LORENTZ
has further shown that if the interaction of matter and light was
regarded to be due to electrons imbedded in the substance all
observations concerning effects of the first order in ft = v/c

PHYSICS AND RELATIVITY igi

(v = velocity of matter, c velocity of light) could be explained,
in particular the fact that no first order effect of the movement of
matter could be discovered by an observer taking part in the
motion. But there were some very accurate experiments such as
that performed by MICHELSON first in 1881 in Potsdam, and repeated
with higher accuracy in America in 1887 by MICHELSON and
MORLEY, which showed that no effect of the earth motion could be
found even to the second order in /?. To explain this FITZGERALD
invented in 1892 the contraction hypothesis, which was at once
taken up by LORENTZ and included in his system. Thus LORENTZ
obtained a set of field equations for moving bodies which was in
agreement with all known observations; it was relativistic invariant
for processes in empty space, and approximately invariant (up to
terms of ist order in /?) for material bodies. Still LORENTZ stuck
to his aether at rest and the traditional absolute time. I shall return
to this point presently. When HENRI POINCAR took up this
investigation, he went a step further. In regard to his work I refer
to the excellent book by Sir EDMUND WHITTAKER, A History of the
Theories of Aether and Electricity, which was already in use as a guide
in my student times. It has now been completely re-written.
The second volume of the new edition deals with 'The Modern
Theories, 1900-1926'; there you can find quotations from POIN-
CARi's papers, some of which I have looked up in the original.
They show that as early as 1899 he regarded it as very probable
that absolute motion is indetectable in principle and that no
aether exists. He formulated the same ideas in a more precise form,
though without any mathematics, in a lecture given in 1904 to a
Congress of Arts and Science at St. Louis, U.S.A., and he predicted
the rise of a new mechanics which will be characterized above all
by the rule, that no velocity can exceed the velocity of light.

WHITTAKER was so impressed by these statements that he gave
to the relevant chapter in his book the title 'The Relativity Theory
of Poincare and Lorentz 3 . EINSTEIN'S contributions appear there as
being of minor importance.

I have tried to form an opinion about this question from my own
recollections and with the help of a few publications available to me.

In the happy years before the first World War the Academy of
Gottingen had a considerable fund, called the Wolfskehl-Stiftung
(W.-Foundation) which was given originally with the direction to
award a prize of 100,000 Marks for the proof of FERMAT'S cele-
brated 'Great Theorem'. Hundreds of letters, or even just post-
cards, arrived every year claiming to contain the solution, and
the mathematicians were kept busy to discover the error. The

iga PHYSICS AND RELATIVITY

futility of this process became so annoying that it was decided to
use the money for other more useful purposes, namely to invite
distinguished scholars to lecture on current scientific problems.
One of these series of lectures was given by HENRI POINCAR&,
April 22nd-28th 1909, and has been published as a book by
Teubner in 1910. I have attended these PoiNCAR^-Festspiele
(P.-Festival), as we called it, and now refreshed my memory by
looking through the book. The first five lectures dealt with purely
mathematical problems; the sixth lecture had the title e La
mecanique nouvelle'. It is a popular account of the theory of
relativity without any formulae and with very few quotations.
EINSTEIN and MINKOWSKI are not mentioned at all, only MICHELSON,
ABRAHAM and LORENTZ. But the reasoning used by POINCAR!
was just that, which EINSTEIN introduced in his first paper of
1905, of which I shall speak presently. Does this mean that POIN-
CAR& knew all this before EINSTEIN? It is possible, but the strange
thing is that this Lecture definitely gives you the impression that
he is recording LORENTZ' work.

On the other hand LORENTZ himself has never claimed to be the
author of the principle of relativity. The year after POINCAR'S
visit to Gottingen we had the LoRENTZ-Festspiele. I, at the time a
young Privatdocent, was appointed temporary assistant to the
distinguished guest and charged with taking notes of the lectures
and preparing them for publication. Thus I was privileged with
having daily discussions with LORENTZ. The lectures have appeared
in Physikalische %eitschrift (vol. 11, 1910, p. 1234). The second
lecture begins with the words: 'Das EiNSTEiNsche Relativitats-
prinzip hier in Gottingen zu besprechen, wo MINKOWSKI gewirkt
hat, erscheint mir eine besonders willkommene Aufgabe 3 . 'To
discuss EINSTEIN'S Principle of Relativity here in Gottingen where
MINKOWSKI has taught seems to me a particularly welcome task. 6
This suffices to show that LORENTZ himself regarded EINSTEIN as
the discoverer of the principle of relativity. On the same page and
also in the following sections are other remarks which reveal
LORENTZ' reluctance to abandon the ideas of absolute space and
time. When I visited LORENTZ a few years before his death, his
scepticism had not changed.

I have told you all these details because they illuminate the
scientific scene of 50 years ago, not because I think that the question
of priority is of great importance.

May I now return to my own struggle with the relativity problem.
After having graduated Dr.phil. in Gottingen I went in 1907 to
Cambridge to learn something about the electron at the source.

PHYSICS AND RELATIVITY 1 93

J. J. THOMSON'S lectures were very stimulating indeed; he showed
brilliant experiments. But LARMOR'S theoretical course did not
help me very much; I found it very hard to understand his Irish
dialect, and what I understood seemed to me not on the level of
MINKOWSKI'S ideas. I then returned to my home city Breslau,
and there at last I heard the name of EINSTEIN and read his papers.
I was working at that time on a relativistic problem, which was an
offspring of MINKOWSKI'S seminar, and talked about it to my friends.
One of them, STANISLAUS LORIA, a young Pole, directed my attention
to EINSTEIN'S articles, and thus I read them. Although I was quite
familiar with the relativistic idea and the Lorentz transformations,
EINSTEIN'S reasoning was a revelation to me.

Many of you may have looked up his paper 'Zur Elektrodynamik
bewegter Korper' in Annalen der Physik (4), vol. 17, p. 811, 1905,
and you will have noticed some peculiarities. The striking point is
that it contains not a single reference to previous literature. It
gives you the impression of quite a new venture. But that is, of
course, as I have tried to explain, not true. We have EINSTEIN'S
own testimony. Dr. CARL SEELIG, who has published a most
charming book on Einstein und die Schweiz asked EINSTEIN which
scientific literature had contributed most to his ideas on relativity
during his period in Bern, and received an answer on February igth
of this year which he published in the Technische Rundschau (N. 20,
47. Jahrgang, Bern 6. Mai 1955); EINSTEIN wrote:

'Es ist zweifellos, daB die spezielle Relativitatstheorie, wenn wir ihre
Entwicklung riickschauend betrachten, im Jahre 1905 reif zur
Entdeckung war. LORENTZ hatte schon erkannt, daB fur die Analyse
der MAXWELLSchen Gleichungen die spater nach ihm benannte
Transformation wesentlich sei, und POINCARE hat diese Erkenntnis
noch vertieft. Was mich betrifft, so kannte ich nur LORENTZ
bedeutendes Werk von 1895 "La theorie electromagnetique de
MAXWELL" und "Versuch einer Theorie der elektrischen und optischen
Erscheinungenin bewegten Korpern" aber nicht LORENTZ*, spatere
Arbeiten, und auch nicht die daran anschlieBende Untersuchung
von POINGARE. In diesem Sinne war meine Arbeit von 1905 selb-
standig.

'Was dabei neu war, war die Erkenntnis, daB die Bedeutung der
Lorentztransformation iiber den Zusammenhang mit den MAXWELL-
schen Gleichungen hinausging und das Wesen von Raum und Zeit
im allgemeinen betraf. Auch war die Einsicht neu, daB die "Lorentz-
Invarianz" eine allgemeine Bedingung sei fur jede physikalische
Theorie. Das war fiir mich von besonderer Wichtigkeit, weil ich
schon fruher erkannt hatte, daB die MAXWELLsche Theorie die Mikro-
struktur der Strahlung nicht darstelle und deshalb nicht allgemein
haltbar sei .'

194 PHYSICS AND RELATIVITY

Translated:

'There is no doubt, that the special theory of relativity, if we regard its
development in retrospect, was ripe for discovery in 1905. LORENTZ
had already observed that for the analysis of MAXWELL'S equations
the transformations which later were known by his name are essential,
and POINCARE had even penetrated deeper into these connections.
Concerning myself, I knew only LORENTZ' important work of 1895
(the two papers quoted above in the German text) but not LORENTZ'
later work, nor the consecutive investigations by POINCAR. In this
sense my work of 1905 was independent. The new feature of it was
the realization of the fact that the bearing of the LORENTZ transforma-
tion transcended its connection with MAXWELL'S equations and
was concerned with the nature of space and time in general. A further
new result was that the "Lorentz invariance" is a general condition
for any physical theory. This was for me of particular importance
because I had already previously found that MAXWELL'S theory did
not account for the micro-structure of radiation and could therefore
have no general validity .'

This, I think, makes the situation perfectly clear. The last
sentence of this letter is of particular importance. For it shows
that EINSTEIN'S papers of 1905 on relativity and on the light quantum
were not disconnected. He believed already then that MAXWELL'S
equations were only approximately true, that the actual behaviour
of light was more complicated and ought to be described in terms
of light quanta (or photons, as we say to-day), but that the principle
of relativity was more general and should be founded on con-
siderations which would be still valid when MAXWELL'S equations
had to be discarded and replaced by a new theory of the fine
structure of light (our present quantum electrodynamics).

The second peculiar feature of this first relativity paper by
EINSTEIN is his point of departure, the empirical facts on which
he built his theory. It is of surprising simplicity. He says that the
usual formulation of the law of induction contains an asymmetry
which is artificial, and does not correspond to facts. According
to observation, the current induced depends only on the relative
motion of the conducting wire and the magnet, while the usual
theory explains the effect in quite different terms according to
whether the wire is at rest and the magnet moving or vice versa.
Then there follows a short sentence referring to the fact that all
attempts to discover experimentally the movement of the earth
through the aether have failed. It gives you the impression that
MICHELSON'S experiment was not so important after all, and
that EINSTEIN would have arrived at his relativity principle in any
case.

PHYSICS AND RELATIVITY 195

This principle together with the postulate that the velocity of
light is constant, independent of the system of reference, are the
only assumptions from which the whole theory is derived on a few
pages. The first step is the demonstration that absolute simultaneity
of two events at different places has no physical meaning. Then
relative simultaneity is defined by setting the clocks at different
places in a system of reference in such a way that a light signal
needs the same time either way between two of them. This definition
leads directly to the Lorentz transformations and all their conse-
quences: the Lorentz-Fitzgerald contraction, the time dilation,
the addition theorem of velocities, the transformation law for the
electromagnetic field components in vacuum, the Doppler principle,
the aberration effect, the transformation law for energy, the equa-
tions of motion for an electron and the formulae for the longitudinal
and transversal mass as functions of the velocity.

But for me and many others the exciting feature of this
paper was not so much its simplicity and completeness, but the
audacity of challenging ISAAC NEWTON'S established philosophy, the
traditional concepts of space and time. That distinguishes EINSTEIN'S
work from his predecessors and gives us the right to speak of EINSTEIN'S
theory of relativity, in spite of WHITTAKER'S different opinion.

EINSTEIN'S second paper on relativity '1st die Tragheit eines
Korpers von seinem Energieinhalt abhangig?' (Ann. d. Phys. (4),
vol. 1 8, 1905, p. 639) contains on three pages a proof of the cele-
brated formula E = me 2 expressing the equivalence of mass and
energy, which has turned out to be of fundamental importance in
nuclear physics, for the understanding of the structure of matter
and of the source of stellar energy as well, and for the technical
exploitation of nuclear energy, for bad or good. This paper also
has become the object of priority disputes. In fact, the formula
had been known for special cases; for instance the Austrian physicist
F. HASENOHRL had shown already in 1904 that electromagnetic
radiation enclosed in a vessel produced an increase of its resistance
to acceleration, i.e. its mass, proportional to the radiation energy.
HASENOHRL was killed in the first world war and could not object
when his name was later misused to discredit EINSTEIN'S discovery.
However, I shall not enter into an account of this sordid story.
I have mentioned these matters only to make it clear that special
relativity was, after all, not a one-man discovery. EINSTEIN'S work
was the keystone to an arch which LORENTZ, POINCARE and others
had built and which was to carry the structure erected by
MINKOWSKI. I think it wrong to forget these other men, as it can
be found in many books. Even PHILIPP FRANK'S excellent biography

ig6 PHYSICS AND RELATIVITY

Einstein, Sein Leben und seine %eit, cannot be acquitted of this reproach,
e.g. when he says (in Chap. 3, No. 6 of the German edition) that
nobody before EINSTEIN had ever considered a new type of mech-
anical law in which the velocity of light plays a prominent part.
Both POINCARE and LORENTZ have been aware of this, and the
relativistic expression for the mass (which contains c) has rightly
been called LORENTZ' formula.

To-day this formula is taken so much for granted that you can
hardly imagine the acerbity of the controversies which raged around
it. In 1901 W. KAUFMANN in Gottingen had by an investigation of
the electromagnetic deflection of fast cathode rays first established
the fact that the mass of the electron depends on its velocity.
MAX ABRAHAM, whom I have mentioned already, took up this
challenge and showed that the electromagnetic mass, as introduced
by J- J- THOMSON, i.e. the self-energy of the electron's own field,
properly developed for high velocities did indeed depend on velocity.
He assumed the electron to be a rigid sphere; but later he also
modified his theory by taking account of the Lorentz-Fitzgerald
contraction, and obtained exactly the formula which Lorentz had
already found by a simpler reasoning. As a matter of fact, the
velocity dependence of energy and of mass has nothing at all
to do with the structure of the body considered, but is a general
relativistic effect. Before this became clear, many theoreticians
wrote voluminous, not to say monstrous, papers on the electro-
magnetic self-energy of the rigid electron G. HERGLOTZ, P.
HERTZ, A. SOMMERFELD, and others. My first scientific attempt was
also in this direction; however, I did not assume the electron to be
rigid in the classical sense, but tried to define relativistic rigidity
by generalizing the Lorentz electron for accelerated motion, with
the help of the methods I had learned from MINKOWSKI.

To-day all these efforts appear rather wasted; quantum theory
has shifted the point of view, and at present the tendency is to cir-
cumvent the problem of self-energy rather than to solve it. But
one day it will return to the centre of the scene.

MINKOWSKI published his paper 'Die Grundlagen fur die elektro-
magnetischen Vorgange in bewegten Korpern' in 1 907. It contained
the systematic presentation of his formal unification of space and
time into a four-dimensional 'world' with a pseudo-euclidean geo-
metry, for which a vector- and tensor calculus is developed. This
calculus, with some modifications, soon became the standard method
of all relativistic investigations. Moreover, MINKOWSKI'S paper
contained important new results : a set of equations for the electro-
magnetic field in moving material bodies which is exactly invariant

PHYSICS AND RELATIVITY ig7

with respect to LORENTZ transformation, not only a first approxi-
mation as LORENTZ' slightly different equations; further a new
approach to the mechanical equations of motion.

In the beginning of 1908 I had the audacity to send my manu-
script on the electron to MINKOWSKI, and he was kind enough to
answer. On September sist of the same year I listened at Cologne
to his famous lecture 'Raum und Zeit', in which he explained his
ideas in popular form to the members of the Naturforscher-Ver-
sammlung. He invited me to come to Gottingen and to join him
in further work. So I did; but alas, after a few weeks our colla-
boration ended through MINKOWSKI'S sudden death. It fell to me
to sift his unpublished papers, one of which I succeeded to recon-
struct and to publish.

My first meeting with EINSTEIN happened in the following year,
1909, at the Naturforscher-Versammlung in Salzburg. There
EINSTEIN gave a lecture with the title 'Dber die neueren Umwand-
lungen, welche unsere Anschauungen iiber die Natur des Lichtes
erfahren haben', which means obviously the introduction of the
light quantum. I also gave a talk 'Die Dynamik des Elektrons im
System des Relativitatsprinzips 5 , This seems to me rather amusing:
EINSTEIN had already proceeded beyond special relativity which
he left to minor prophets., while he himself pondered about the
new riddles arising from the quantum structure of light, and of
course about gravitation and general relativity which at that time
was not ripe for general discussion.

From this time on I saw EINSTEIN occasionally at conferences
and exchanged a few letters with him. He became professor at the
University of Zurich in 1909, then at Prague in 1910 and returned
to Zurich, as professor at the Polytechnicum in 1912. Already in
the following year he went to Berlin, where the Prussian Academy
had offered him a special chair, vacated by the death of VAN't
HOFF, with no teaching obligations, and with other privileges. This
invitation was mainly due to the efforts of MAX PLANCK who was
deeply interested in relativity and had contributed important papers
on relativistic mechanics and thermodynamics. Two years later,
in spring 1915, I was also called to Berlin by PLANCK, to assist him
in his teaching. The following four years have been amongst the
most memorable of my life, not because the first World War was
raging with all its sorrows, excitements, privations and indignities,
but because I was near to PLANCK and EINSTEIN.

It was the only period when I saw EINSTEIN very frequently, at
times almost daily, and when I could watch the working of his
mind and learn his ideas on physics and on many other subjects.

198 PHYSICS AND RELATIVITY

It was the time when general relativity was finally formulated.
Now this was, in contrast to the special theory, a real one-man
work. It began with a paper published as early as December,
1907, which contains the principle of equivalence, the only empirical
pillar on which the whole imposing structure of general relativity
was built.

When speaking of the physical facts which EINSTEIN used in 1905
for his special relativity I said that it was the law of electromagnetic
induction which seemed to have guided EINSTEIN more than even
MICHELSON'S experiment. Now the induction law was at that time
about 70 years old (FARADAY discovered it in 1834), everybody had
known all along that the effect depended only on relative motion,
but nobody had taken offence at the theory not accounting for this
circumstance.

Now the case of the equivalence principle is very similar, only
that the critical empirical fact has been known by everybody far
longer, namely about 250 years. GALILEO had found that all
bodies move with the same acceleration under terrestrial gravity,
and NEWTON generalized this for the mutual gravitational attraction
of celestial bodies. This fact, namely, that the inertial and the
gravitational mass are equal, was taken as a peculiar property of
NEWTON'S force, and nobody seems to have pondered about it.

Special relativity had restored the special r61e and the equivalence
of the inertial systems of Newtonian mechanics for the whole of
physics; absolute motion was indetectable as long as no accelera-
tions occurred. But the inertia effects, the centrifugal forces and
corresponding electromagnetic phenomena, which appear in accel-
erated, for instance rotating, systems could be described only in
terms of absolute space. This seemed to be intolerable to EINSTEIN.
Brooding over it, he noticed that the equality of inertial and gravi-
tational mass implied that an observer in a closed box could not
decide whether a non-uniformity of the motion of a body in the box
was due to an acceleration of the whole box or to an external gravi-
tational field. This gave him the clue for general relativity.
EINSTEIN postulated that this equivalence should hold as a general
principle for all natural phenomena, not only mechanical motion.
Thus he arrived in 1911 at the conclusion that a beam of light
must be bent in a gravitational field and suggested at once that his
simple formula of deflexion could be experimentally checked by
observing the position of fixed stars near the sun during a total
eclipse.

The actual development of the theory was a tremendous task,
for a new branch of mathematics, quite unfamiliar to physicists,

PHYSICS AND RELATIVITY igg

had to be used. Some more conservative physicists, ABRAHAM,
MIE, NORDSTROM and others tried to develop from EINSTEIN'S
equivalence principle a coherent scalar theory of the gravitational
field, with little success. EINSTEIN himself was the only one who
discovered the right mathematical tool in RIEMANN'S geometry, as
extended by RICCI and LEVi-CrvrrA, and he found in his old friend
MARCEL GROSSMANN a skilful collaborator. But it took several years,
until 1915, to finish this work.

I remember that on my honeymoon in 1913 I had in my luggage
some reprints of EINSTEIN'S papers which absorbed my attention
for hours, much to the annoyance of my bride. These papers seemed
to me fascinating, but difficult and almost frightening. When I
met EINSTEIN in Berlin in 1915 the theory was much improved and
crowned by the explanation of the anomaly of the perihelion of
Mercury, discovered by LEVERRIER. I learned it not only from the
publications but from numerous discussions with EINSTEIN which
had the effect that I decided never to attempt any work in this
field. The foundation of general relativity appeared to me then,
and it still does, the greatest feat of human thinking about Nature,
the most amazing combination of philosophical penetration, physical
intuition and mathematical skill. But its connections with experience
were slender. It appealed to me like a great work of art, to be
enjoyed and admired from a distance.

According to my interpretation of the title of this lecture I shall
not enter into a discussion of the empirical confirmation of the
special and the general theory of relativity, as I am no expert, and
as others have spoken of it already I shall only just mention the
most striking events.

In 1915 SOMMERFELD'S relativistic theory of the fine structure of
the hydrogen lines was published. It is based on the mathematical
result, that the dependence of mass on velocity produces a pre-
cession of the perihelion of the elliptic orbit. It is quite interesting
that POINCARE had already considered this effect to explain
LEVERRJER'S anomaly in the motion of the planet Mercury; a
remark about this is contained in POINGARE'S lecture in Gottingen
quoted before. The result was of course negative, as the velocity of
Mercury is much too small compared with that of light. It is
different with the electron moving around a nucleus and, in com-
bination with the quantization laws of BOHR and SOMMERFELD,
this led to the explanation of the splitting of the hydrogen lines.

The modern version of the theory of the hydrogen spectrum is
based on DIRAC'S relativistic wave equation and has recently been
much refined with the help of quantum electrodynamics.

20O PHYSICS AND RELATIVITY

Another striking result of relativity combined with EINSTEIN'S
idea of light quanta is the theory of the Compton effect.

The time dilation effect was directly confirmed as the transversal
DOPPLER effect on hydrogen canal rays in 1938 by IVES and
STTEVELL, and with higher accuracy in 1939 by RUCHARDT and
OTTING. It plays an important part in the modern research on
mesons in cosmic rays where the observed lifetime of a meson may
be a hundred times as large as the intrinsic one in consequence of
the large velocities.

At present special relativity is taken for granted, the whole
of atomic physics is so merged with it, so soaked in it, that it would
be quite meaningless to pick out particular effects as confirmations
of EINSTEIN'S theory. The situation in general relativity is different;
all the three effects predicted by EINSTEIN exist, but the question of
quantitative agreement between the theory and observation is
still under discussion. However, the importance of general relativity
lies in the revolution which it has produced in cosmology. It started
in 1917 when EINSTEIN generalized his field equations by adding
the so-called cosmological term and showed that a solution exists
representing a closed universe. This suggestion of a finite, but
unbounded, space is one of the very greatest ideas about the nature
of the world which ever has been conceived. It solved the mysterious
fact why the system of stars did not disperse and thin out, which it
would do if space were infinite; it gave a physical meaning to
MACK'S principle which postulated that the law of inertia should
not be regarded as a property of empty space but as an effect of
the total system of stars, and it opened the way to the modern con-
cept of the expanding universe. Here general relativity found again
contact with observation through the work of the astronomers
SHAPLEY, HUBBLE and many others. To-day cosmology is an
extensive science which has produced innumerable publications and
books, of which I know little. Thus I am compelled to omit just
that aspect of EINSTEIN'S work which may be regarded as his
greatest achievement.

May I, instead, tell you something about my personal relations
with EINSTEIN in those bygone days and about the divergence of
opinion which arose in the end between us in regard to the ultimate
principles of physics.

The discussions which we had in Berlin ranged far beyond rela-
tivity, and even beyond physics at large. As the first world war
was going on politics played of course a central part. But much
as I would like to speak about these things I have to restrict myself
to physics.

PHYSICS AND RELATIVITY 2OI

EINSTEIN was at that time working with DE HAAS on experiments
about the so-called gyromagnetic effect, which proved the existence
of AMPERE'S molecular currents. He was also deeply interested in
quantum theory but worried by its paradoxes.

In 1919 I became v. LAUE'S successor at Frankfurt, and my
companionship with EINSTEIN ceased. But we visited one another
often and had a lively correspondence, of which I shall give you a
few examples. It was the time when EINSTEIN suddenly became
world famous, and his theory as well as his personality the object
of fanatical controversy.

Just before the war a German expedition had gone to Russia to
investigate EINSTEIN'S prediction of the deflexion of light by the sun
during an eclipse; they were stopped by the outbreak of hostilities,
and became prisoners of war. Now after the war two British
expeditions went out for the same purpose, under the direction of
Sir ARTHUR EDDINGTON, and they were successful. It is quite
impossible to describe the stir which this event produced in the
whole world. EINSTEIN became at once the most famous and
popular figure, the man who had broken through the wall of hatred
and united the scientists to a common effort, the man who had
replaced ISAAC NEWTON'S system of the world by another and
better one. But at the same time an opposition, which had already
been apparent while I was in Berlin, grew under the leadership of
PHILIPP LENARD and JOHANNES STARK. It was springing from the
most absurd mixture of scientific conservatism and prejudice with
racial and political emotions, due to EINSTEIN'S Jewish descent and
pacifistic, antimilitaristic convictions. Here a few samples from
EINSTEIN'S letters; one of June 4th 1919 begins with physics:

*. . . Die Quantentheorie lost bei rnir ganz ahnliche Empfindungen
axis wie bei Ihnen. Man miiBte sich eigentlich der Erfolge schamen,
weil sie nach dem jesuitischen Grundsatze gewonnen sind: "Die
eine Hand darf nicht wissen, was die andre tut . . .".'

'. . , The quantum theory provokes in me quite similar sensations as
in you. One ought really to be ashamed of the successes, as they are
obtained with the help of the Jesuitic rule: "One hand must not
know what the other does".'

and then, a few lines below, he continues about politics:

'. . . Darf ein hartgesottener -X"-Bruder und Determinist mit thranen-
feuchten Augen sagen, daB er den Glauben an die Menschen verloren
hat? Gerade das triebhafte Verhalten der Menschen von heute in
politischen Dingen ist geeignet, den Glauben an den Determinismus
recht lebendig zu machen . . .'

2O2 PHYSICS AND RELATIVITY

*. . . Can a hardboiled -ST-brother (= mathematician; we used the
expression "ixen", "to AT", for "calculating") say with tears in his
eyes that he has lost his faith in the human race? Just the instinctive
behaviour of contemporary people in political affairs is apt to revive
the belief in determinism . . .'

You see that his deterministic philosophy which later created a
gulf between him and the majority of physicists was not restricted
to science but extended to human affairs as well.

At this time the inflation in Germany began to become serious.
In my department STERN and GERLAGH were preparing their well-
known experiments, but hampered by the lack of funds. I decided
to give a series of popular lectures on relativity with an entrance
fee, using the general craze for information about this subject to
raise funds for our researches. The plan was successful, the lectures
were crowded, and when they appeared as a book three editions
were quickly sold. EINSTEIN acknowledged my efforts by offering
me the friendly 'Du J instead of the formal e Sie* in a letter of Novem-
ber gth 1919, which also contains some suggestion how the Jews
should react to the antisemitic drive going on:

'Also von jetzt ab soil Du gesagt werden unter uns, wenn Du es
erlaubst . . . Ich wiirde es fiir vernunftig halten, wenn die Juden
selbst Geld sammelten, um jiidischen Forschern auBerhalb der Univer-
sitaten Unterstiitzung und Lehrgelegenheit zu bieten . . .'

'Well, from now on the "Thou" shall be used between us, if thou
agreest ... I should think it reasonable if the Jews themselves would
collect money in order to give Jewish scholars financial support and
teaching facility outside the universities . . .'

There appeared attacks against EINSTEIN by well known scientists
and philosophers in the Frankfurter ^eitung which aroused my
pugnacity. I answered in a rather sharp article. EINSTEIN seems
to have been pleased with it for he wrote on December 9th 1919:

'Dein ausgezeichneter Artikel in der Frankfurter Zeitwg hat mich sehr
gefreut. Nun aber wirst Du, gerade wie ich, wenn auch in schwa-
cherem Masstab, von Presse- und sonstigem Gelichter verfolgt Bei
mir ist es so arg, dafi ich kaum mehr schnaufen, geschweige zu
vernunftiger Arbeit kommen kann . . .'

'Your excellent article in the Frankfurter %eitung has given me great
pleasure. Now you as well as I will be persecuted by gangs of pressmen
and others though to a smaller degree. With me it is so bad that I
can hardly breathe any more, to say nothing of doing reasonable
work . . .'

PHYSICS AND RELATIVITY 2O3

And about a year later (September gth 1920) :

'. . . Wie bei dem Mann im Marchen alles zu Gold wurde, was er
beriihrte, so wird bei mir alles zum Zeitungsgeschrei: Suum
cuique . . .*

e . . . Just as with the man in the fairy tale everything he touched was
transformed into gold, with me everything becomes newspaper noise.
Suum cuique . . .'

If you are interested in that curious period when a whole world
was excited about a physical theory which nobody understood,
and when everywhere people were split into pro- and contra-EiN-
STEIN factions you can find an excellent account in the biography
by PHILIPP FRANK quoted before.

However, scientific problems regained their proper place in our
correspondence. In the same year (March 3rd 1920) EINSTEIN
wrote:

*Ich brute in meiner freien Zeit immer iiber dem Quantenproblem
vom Standpunkte der Relativitat. Ich glaube nicht, dafi die Theorie
das Kontinuum wird entbehren konnen. Es will mir aber nicht
gelingen, meiner Lieblingsidee, die Quantentheorie aus einer Uber-
bestimmung durch Differentialgleichungen zu verstehen, greifbare
Gestalt zu geben . . .'

'I always brood in my free time about the quantum problem from
the standpoint of relativity. I do not think that the theory will have to
discard the continuum. But I was unsuccessful, so far, to give tangible
shape to my favourite idea, to understand the quantum theory with
the help of differential equations by using conditions of over-deter-
mination . . .'

Already at that time we discussed whether quantum theory could
be reconciled with causality. Here a sentence from EINSTEIN'S
letter of January syth 1920:

*. . . Das mit der Kausalitat plagt mich auch viel. 1st die quanten-
hafte Licht- Absorption und -Emission wohl jemals im Sinne der
vollstandigen Kausalitatsforderung erfassbar oder bleibt ein statistischer
Rest? Ich muss gestehen, dass mir da der Mut einer "Oberzeugung
fehlt. Ich verzichte aber sehr, sehr tmgern auf volktandige Kausal-
itat . . .'

'That question of causality worries me also a lot. Will the quantum
absorption and emission of light ever be grasped in the sense of complete
causality, or will there remain a statistical residue? I have to confess,
that I lack the courage of a conviction. However I should be very,
very loath to abandon complete causality . . ."

204 PHYSICS AND RELATIVITY

From that time on our scientific ways parted more and more. I
went to Gottingen and came in contact with NIELS BOHR, PAULI
and HEISENBERG. When in 1927 quantum mechanics was deve-
loped, I hoped of course that EINSTEIN would agree, but was
disappointed. Here a quotation from one of his letters (December
1 2th 1926):

*. . . Die Quantenmechanik ist sehr achtunggebietend. Aber
eine innere Stimme sagt mir, dass das doch nicht der wahre Jakob
ist. Die Theorie liefert viel, aber dem Geheimnis des Alten bringt
sie uns kaum naher. Jedenfalls bin ich iiberzeugt, daB der nicht
wiirfelt . . . Ich plage mich damit herum, die Bewegungsgleichungen
von als Singularitaten aufgefassten materiellen Punkten aus den
DifFerentialgleichungen der allgemeinen Relativitat abzuleiten . . .*

'The quantum mechanics is very imposing. But an inner voice tells
me that it is still not the true Jacob [a German colloquialism.]. The
theory yields much, but it hardly brings us nearer to the secret of the
Old One. In any case I am convinced that he does not throw dice . .
I am toiling at deriving the equations of motion of material particles
regarded as singularities from the differential equations of general
relativity . . .'

The last sentence refers to a paper which was finished much
later at Princeton in collaboration with BENESH HOFFMANN and
LEOPOLD INFELD, EINSTEIN'S last great contribution to relativity.
The assumption made in the original theory, that a free particle
(e.g. a celestial body) moves on a geodesic turned out to be un-
necessary, it could be derived from the field equations by a subtle
procedure of successive approximations. These very deep and
important investigations have been further developed by FOGK
and INFELD.

The first part of the letter quoted refers to EINSTEIN'S refusal to
accept statistical laws in physics as final; he speaks of the dice-
playing God, an expression which he has used later very often in
discussion and letters.

During the last period of his life in Princeton he concentrated
all his powers and energies on developing a new foundation of physics
in conformity with his fundamental philosophical convictions,
namely that it must be possible to think of the external world as
existing independently of the observing subject, and that the laws
governing this objective world are strictly causal, in the sense of
deterministic. This was the aim of his unified field theories, of
which he published several versions, always hoping that the quantum
principles would in the end turn out to be a consequence of his
field equations.

PHYSICS AND RELATIVITY 2O5

I cannot say much about these attempts, as right from the begin-
ning I just did not believe in their success and therefore did not
study his difficult papers with sufficient care. I think that quantum
mechanics has followed up EINSTEIN'S original philosophy, which
led him to tremendous success, more closely than he did himself
in his later period.

What is this lesson we learned from him? He himself has told us
that he learned it from ERNST MACH, and therefore the positivists
have claimed him to be one of them. I do not think this is true,
if positivism is the doctrine that the purpose of science is the
description of interrelation of sense impressions. EINSTEIN'S leading
principle was simply that something of which you could think and
form a concept, but which from its very nature could not be sub-
mitted to an experimental test (like the simultaneity of events at
distant places) has no physical meaning.

The quantum effects showed that this holds for a great many
concepts of atomic physics, but EINSTEIN refused to apply his
criterion to these cases. Thus he rejected the current interpretation
of quantum mechanics, though it follows his own general teaching,
and tried quite a different way, rather remote from experience.
He had achieved his greatest success by relying on just one empirical
fact known to every schoolboy. Yet now he tried to do without
any empirical facts, by pure thinking. He believed in the power
of reason to guess the laws according to which God has built the
world. He was not alone in this conviction. One of the principal
exponents of it was EDDINGTON in his later papers and books. In
1943 I published a pamphlet with the title Experiment and Theory
in Physics (Cambridge University Press) in which I tried to analyse
the situation and to refute EDDINGTON'S claims. I sent a copy to
EINSTEIN and received a very interesting reply which unfortunately
has been lost; but I remember a phrase like this : *Your thundering
against the Hegelism is quite amusing, but I shall continue with
my endeavours to guess God's ways.' A man of EINSTEIN'S greatness
who has achieved so much by thinking, has the right to go to the
limit of the a priori method. Current physics has not followed him;
it has continued to accumulate empirical facts, and to interpret
them in a way which EINSTEIN thoroughly disliked. For him a
potential or a field component was a real natural object which
changed according to definite deterministic laws. Modern physics
operates with wave functions which, in their mathematical behavi-
our, are very similar to classical potentials but do not represent real
objects; they serve for determining the probability of finding real
objects, whether these are particles or electromagnetic potentials,

206 PHYSICS AND RELATIVITY

or other physical quantities. EINSTEIN made many attempts to
prove the inconsistency of this theory with the help of ingenious
examples and models, and NIELS BOHR took infinite trouble to refute
these attacks; he has given a charming report about his discussions
with EINSTEIN in the book Einstein, Philosopher-Scientist (The Library
of Living Philosophers, Vol. 7, p. 199).

I saw EINSTEIN the last time about 1930, and although our
correspondence continued I do not feel competent to speak about
the last phase of EINSTEIN'S life and work. I hope that Professor
PAULI will tell us something about it. I conclude my address by
apologizing that it was so long. But my friendship with EINSTEIN
was one of the greatest experiences of my life, and 'Ex abundantia
enim cordis os loquitur', or in good Scots: 'Neirest the heart,
neirest the mouth'.

DEVELOPMENT AND ESSENCE
OF THE ATOMIC AGE

[Lecture given to a meeting of journalists held at the Protestant Theological
Academy of Loccum Abbey, Niedersachsen, Germany, on- March i8th, I955>
and repeated at several other meetings during the summer of 1955.]

IN following the invitation to speak about the Atomic Age,
its development and essence, I do not think that I am intended
to enlarge upon physical discoveries and their applications to
technological and military ends, but rather on what appears to
me the historical roots of these discoveries and their consequences
upon the destiny of Man. But a scientist like myself has little time
for historical studies; I have to rely on the fact that during my
long life of more than 70 years I have witnessed a section of modern
history and pondered about it. Moreover, I have read or at least
scanned a few books which may be useful for my purpose. For
instance, I remember from my student's days SPENGLER'S Decline
of the West (Untergang des Abendlandes) . I have also read a little
in ARNOLD TOYNBEE'S great work, and listened to some of his
Gifford Lectures given at Edinburgh a few years ago. I mention
these two authors together because both share the opinion that
there are regularities or even laws in human history which can
be revealed by a comparative study of different groups of nations
and civilizations. What I actually know of European history is
essentially due to a book much used at British schools and elementary
University courses because of its admirable style and clarity,
H. A. L. FISHER'S A History of Europe. His standpoint can be seen
from quoting a few lines of his Preface.

'One intellectual excitement has, however, been denied to me. Men
wiser and more learned than I have discerned in history a plot, a
rhythm, a predetermined pattern. These harmonies are concealed
from me. I can see only one emergency following upon another as
wave follows upon wave, only one great fact with respect to which,
since it is unique, there can be no generalizations, only one safe rule
for the historian: that he should recognize in the development of
human destinies the play of the contingent and the unforeseen. This
is not a doctrine of cynicism and despair. The fact of progress is
written plain and large on the page of history; but progress is not a
law of nature. The ground gained by one generation may be lost by
the next. Thoughts of men may flow into the channels which lead to
disaster and barbarism.*

207

208 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

There are apparently two historical schools, one of which believes
that the historical course of events obeys laws and has a meaning,
and another which denies this.

As a scientist I am accustomed to search for regularities and laws
in natural phenomena. I beg your forebearance if I consider also
the problem in hand from this standpoint, yet in quite a different
manner from that used by the two historians mentioned.

The dawn of a new historical age, for instance the transition
from antiquity to the mediaeval period, is obviously not noticed by
those who are alive at that time. Everything goes on without
break, the life of the son is not much different from that of the
father. The division into periods and ages is an invention of the
historians made for the purpose of finding their way in the chaos
of events. Even the beginning of the scientific-technological period
in which we are living was a slow process stretching over more
than a hundred years and hardly noticed by the people of that
day.

At present things appear to be different. During the time of a
few years something new has arrived which is transforming our
lives. This new feature includes simultaneously a horrible threat
and a brilliant hope: the threat of self-destruction of the human
race, the hope of earthly paradise. And this is not a revelation of
religious prophets or of philosophical sages, but these two possi-
bilities are presented to the human race for choosing by science,
the most sober activity of the mind. The threat of destruction in
particular is demonstrated by impressive examples Hiroshima
and Nagasaki which should suffice to convince. But I wish to
say right from the beginning that the atom bombs used there were
children's toys compared with the thermo-nuclear weapons deve-
loped since. I myself have not taken any part in the development
of this chapter of science nuclear physics. But I know enough
of it to say that it is not a question of a simple multiplication of
destructive power, which would lead to the annihilation of a
certain number of unfortunate people, while a much greater
number of more fortunates would escape. It is a radical and
sweeping change of the situation. Already to-day the stock of
A-, H- and U-bombs in the United States and in Russia is probably
sufficient to wipe out mutually all larger cities in both countries,
and presumably in addition all remaining centres of civilization,
since almost all countries are more or less attached to one of the
giant powers. But much worse things are in preparation, perhaps
already available for application: for instance the cobalt bomb
which produces a radioactive dust spreading over wide areas and

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 2O9

killing all living creatures therein. Particularly sinister are the
after-effects of radioactive radiation on generations unborn;
mutations may be induced which lead to a degeneration of the
human race. OTTO HAHN whose discovery of the fission of uranium
has set in motion this development without his participation and
much against his wish recently described the true aspect of the
situation in a radio lecture which has been published and widely
read; I need to add nothing to it. There he has also mentioned
the useful applications of nuclear physics, namely, the generation
of energy, the production of isotopes as instruments in medicine
and technology, and so on. These may indeed become a blessing
in future days, but only if these future days exist. We are standing
at a crossroad as the human race has never met before on her way
through the centuries.

This 'to be or not to be* is, however, only a symptom of a state
of our mental development. We have to ask: What is the deeper
cause of the dilemma in which man has been involved?

The fundamental fact is the discovery that the matter which we
and all things around us are made of is not solid and indestructible,
but unstable, an explosive. We are all sitting, in the true sense of
the word, on a powder barrel. This barrel has, it is true, rather
strong walls, and we needed a few thousand years to drill a hole
into it. To-day we have just got through and may at any moment
blow ourselves sky-high with a match.

This dangerous situation is simply a matter of fact. I shall return
to the scientific facts later and describe them in more learned
terms. But first I want to discuss the question: Would it not have
been possible to let the barrel untouched and to sit peacefully
upon it without caring about its content? Or, without the use of
this metaphor: Could the human race not live and flourish without
investigating into the structure of matter and thus to conjure up
the peril of self-destruction ?

To answer this question one needs a definite philosophical out-
look on history. I am hardly entitled to claim any knowledge
in this field, yet, as I proposed before, permit me to try and tackle
it with the methods of a scientist.

Then the situation appears like this. Man is often defined as the
"thinking animal'. His rise depends on his ability to collect expe-
riences and to act accordingly. Single individuals or groups of
such lead the way, others follow and learn. This was an anonymous
process through centuries; we know nothing of the men of early
ages who invented the first tools and weapons, who learned cattle
breeding and agriculture, who developed the languages and the
art of writing. But we may be sure that there was a permanent

21 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

struggle between the minority of progressively-minded people and
the conservative crowd, as we observe it since written documents
exist. The total number of men is large and increases with each
improvement of the conditions of life. If the percentage of the
gifted remains roughly constant their absolute number grows in the
same rate as the total number of men. Simultaneously with each
technical invention the possibility of new combinations increases.
Hence the situation is similar to that of the calculus of compound
interest: If the interest is added to the capital this increases, and
with it the next instalment of interest, hence again the capital, and
so on ad infinitum. One has what the mathematicians call an exponen-
tial increase.

This is, of course, only correct for the average, it is a statistical
law. I am convinced that the laws of statistics are valid in history
just as for the game of roulette, or in atomic physics, in stellar
astronomy, in genetics and so on namely, in all cases where one
has to do with large numbers. This may be taken as an interpreta-
tion of the meaning of the sentence from FISHER'S History of Europe,
quoted above: 'The fact of progress is written plain and large on
the page of history'. But if he continues, but progress is not a law
of nature' he appears to have applied an obsolete notion of the
essence of natural laws, namely, that they are rigorously causal
and deterministic and permit no exception. We know to-day that
most of the laws of nature are of a statistical kind and permit
deviations; we physicists call these 'fluctuations'.

As this idea is not familiar to everybody allow me to illustrate
it by a simple example. The air which we all breathe seems to be
a thin, continuous substance of uniform density. But investigations
with intricate instruments have shown that actually the air consists
of innumerable molecules (mainly of two kinds, oxygen and hydro-
gen) which fly about and collide with one another. The appearance
of continuity is a consequence of the grossness of our senses which
register only the average behaviour of big numbers of molecules.
But then the question arises: why is the average distribution uniform
in the chaotic dance of molecules? Or in other words, why is there
the same number of molecules in two equal volumes of space?
The answer is that there is never exactly the same number of
molecules in equal volumes, but only approximately, and this is the
consequence of a simple result of statistics, according to which this
approximately uniform distribution has an overwhelming probability
as compared with any others. But there are deviations which can
be observed if the two volumes compared are sufficiently small.
Particles suspended in the air, for instance pollen from plants or

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 211

cigarette smoke, perform tiny irregular zig-zag motions which can
be seen in a microscope; the explanation given by EINSTEIN of
this effect, called Brownian movement, is simply that the number
of air molecules hitting such a tiny, but microscopically visible,
particle in opposite directions is not exactly equal in any short
time interval, hence the particle is pushed about through the
fluctuations of the average recoil. In principle there is no limit
to the size of these fluctuations, but a statistical law makes it
extremely improbable that very large deviations occur. Otherwise
it might happen that the density of the air near to my mouth might
become so small for a few minutes that I would suffocate. I am
not afraid of this because the probability of its occurrence is
immensely small.

I think that uniformity in history is due to the same statistical
law. But ordinary history deals generally with small groups and
short times; then the statistical uniformity does not strike the eye,
but the fluctuations which appear chaotic and senseless. I wonder
whether TOYNBEE'S speculations may not be regarded as an attempt
to discover regularities in the fluctuations.

However that may be, one conclusion from this consideration
seems to me inescapable.

The process of gathering and applying knowledge seen as an
endeavour of the whole human race over long periods of time must
follow the statistical law of exponential increase and cannot be
halted.

On the other hand, if only a restricted space on earth and a
restricted period are considered, say a nation or a group of peoples
in the period of a few hundred years, nothing of that process may
be visible, even a loss of the achievements and a retrogression. But
then the power of the human mind will manifest itself at another
place of the world and at another time.

Let me illustrate this by a few historical reminiscences. The
decisive step on the way to atomic physics was made about 2,500
years ago; I mean the speculations of the Greek school of natural
philosophy, THALES, ANAXIMANDER, ANAXIMENES, especially the
atomists LEUKIPPOS and DEMOGRITOS. They were the first who
thought about Nature without expecting an immediate material
advantage, driven by a pure desire of knowledge. They postulated
the existence of natural laws and tried to reduce the variety of
matter to the play of configuration and motion of invisible, un-
changeable, equal particles. It is not easy to apprehend the immense
superiority of this idea over all conceptions current at that time in
the rest of the world. Together with the grand mathematics of the

212 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

Greeks this idea might have led already at that early period to a
decisive scientific- technological advance had not the social condi-
tions been unfavourable. These Greek gentlemen lived in a world
which venerated the harmony and beauty of body and mind. They
despised manual work which was the task of slaves, and thus they
neglected experiment which cannot be done without soiling one's
hands. Thus no empirical foundation of the ideas was attempted,
nor their technical application, which might have saved the antique
world from the assault of the barbarians.

After the great migration of people the Christian Church erected
a totalitarian system ill-disposed to all innovations. Yet the fire
kindled by the Greeks smouldered under the ashes. It lay hidden
in the books which were kept and copied in many monasteries and
stored in the libraries of Byzantium, and it flared up to a bright
flame through the Arabian scholars who even created essentially
new things in mathematics and astronomy and who guarded the
Greek tradition until the time was ripe. The Byzantians who fled
before the Turks to Italy did bring with their books not only the
knowledge of classical antiquity but also the idea of research. Thus
came the time of discoveries and inventions which secured Europe's
preponderance for a few centuries. A parallel development, per-
haps of even older origin, took place in China. I know little about
it, but there is a new comprehensive book by J. NEEDHAM, well
known biologist at Cambridge, England, which gives a detailed
account of it. During and after the European Renaissance, China
was just in a state of rest or stagnation, and thus it came about that
Europe was ahead for a few centuries. I had enough Chinese, also
Japanese and Indian, students to be convinced that these nations
are in no way inferior to us in scientific talent.

There are two conclusions to be drawn from these considerations.
Firstly, it is quite absurd to believe the crisis in the existence of the
human race, the dawn of the atomic age, might have been avoided,
or the further development of dangerous knowledge might be
inhibited. HITLER has tried to choke what he called 'Jewish
Physics', the Soviets tried the same with Mendelian genetics, both
without any success, to their own detriment.

Secondly, the suddenness of the appearance of the critical situa-
tion is partly an historical accident, but mainly a deception of
perspective distortion. The knowledge of Nature and the power
springing from it are steadily growing, though with fluctuations
and retrogressions, but in the average with the continuously in-
creasing acceleration characteristic for a self-supporting (exponen-
tial) process. Thus the day had necessarily to come when the

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 213

change of the conditions of life produced by this process would be
considerable during one single generation and therefore would
appear as a catastrophe. This impression of a catastrophe is in-
creased by the complications due to the fact that there are peoples
which have not taken part in the technical development and have
to adapt themselves to it without proper preparation.

It is our generation which gathers the harvest sown by the Greek
atomists. The final result of physical research is a confirmation of
their fundamental idea that the material world is essentially com-
posed of equal elementary particles whose configuration and inter-
action produces the variety of phenomena. But this simple des-
cription is, of course, only a crude condensation of an abundance
of experimental results, and becomes, by supplementary features,
in the end very complicated.

Those elementary particles are called nucleons, because by
clotting together they form the atomic nucleus. The chemical
atoms are neither invisible (as the name indicates), nor all identical
for a definite chemical element, as believed during the last century.
This is a consequence of the fact that a nucleon may be either
electrically neutral then it is called neutron or may carry a
positive elementary charge then it is called proton. The chemical
atoms consist of a nucleus which is an extremely dense agglomera-
tion of neutrons and protons (hence it is positively charged), and
an extended cloud of negative electric particles, called electrons,
surrounding the nucleus. The electron has a very small mass
compared with the nucleon, but the same charge as the proton,
with the opposite sign. The number of electrons in the cloud is
equal to that of the protons in the nucleus so that the whole atom
is electrically neutral. The electronic cloud determines the chemical
and most of the physical properties of the atom. Atoms which
have the same number of protons and therefore the same number of
electrons in the cloud are chemically, and in most respects also
physically, indistinguishable, even if the number of neutrons in the
nucleus may differ. Such almost identical atoms, which differ only
by the number of neutrons, i.e. by their mass (weight) are called
isotopes.

The elements of ordinary chemistry and physics are mixtures of
isotopes. The laws which govern the structure of the electronic
cloud are known; the current research in this field is not concerned
with the discovery of new principles but with the treatment of
cases of increasing complexity. The laws governing the structure
and behaviour of the nuclei are not so well explored. However, it
is perfectly certain that some of the most general physical laws are

214 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

valid there too, and with their help far-reaching conclusions can
be drawn.

The most important of these laws is that which formulates the
equivalence of mass (M) and energy (), expressed by the frequently
quoted formula E = Me 2 where c is the velocity of light. Its general
derivation was given by EINSTEIN, exactly 50 years ago, with the
help of relativistic reasoning, long before there existed any possibility
of an experimental test. The number c is, in ordinary units, centi-
metres per second, very large, a 3 with 10 zeros behind it; hence
c* == c x c is extremely large, a 9 with 20. zeros. Therefore the
change of mass (M = Ejc*} is excessively small for all ordinary
chemical and physical energy exchanges. In principle a clock
becomes a little heavier when wound up, but that is absolutely
immeasurable. The situation is different for nuclear transformations
where large energies are exchanged.

A piece of a wall consisting of 100 equal bricks without mortar
has a weight exactly 100 times that of a single brick; if there is
mortar the weight is correspondingly higher. The same holds
roughly for nucleons: a nucleus containing 100 nucleons is about
100 times as heavy as a single nucleon. Yet only approximately:
there are deviations, hence there must be a kind of mortar. Now
strangely enough this mortar appears to have a negative weight:
the nucleus is lighter than the sum of its constituents. Namely,
according to EINSTEIN, the mortar is the binding energy which is
lost when the parts are combined. These 'mass-defects' are con-
siderable, hence the corresponding energies enormously large.

The lightest element, hydrogen, consists of one isotope, the single
proton. (There is also a hydrogen isotope with one additional
neutron deuteron and one with two neutrons triton.) The
next element, helium, consists mainly of an isotope having 2 protons
and 2 neutrons. When these agglomerate energy is liberated, very
much energy. The process does not occur spontaneously because
there is an initial obstacle against the combination of the 4 particles,
some energy has to be spent. The situation is like that of a water
barrage, the gates of which have to be wound up before the water
in the reservoir can stream out and do work. The same holds for
the consecutive elements; they are instable and would combine
unless there were barrages, fortunately very strong barrages, to
keep them apart. This is the case in the series of elements up to
the middle of the whole system, about the place of iron; from there
on the situation is reversed, each nucleus has the tendency to split
and is only prohibited to do so by a barrage. The last of the elements
found in Nature, uranium, has the weakest barrage, and it was this

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 215

one which was first broken in the experiments by HAHN and his
collaborator STRASSMANN in 1938.

The way from these delicate laboratory experiments to the first
uranium reactor (or pile) which was built in Chicago by ENRICO
FERMI in 19423 was long and demanded an enormous amount of
ingenuity, courage, skill, organization and money. The decisive
discovery was that the fission of a uranium nucleus produced by
the collision with a neutron is accompanied by the emission of
several neutrons, and that the process could be directed in such a
way that a sufficient number of these could be prevented from
escaping or being lost by collisions with impurities as to produce an
avalanche of new fissions, a self-containing reaction. To begin
with, nobody could predict the outcome, but Nature has arranged
it in this manner, hence it was discovered by Man as soon as the
means were available. That they were available was a historical
accident, a consequence of the great war. The technological process
to produce a bomb until its explosion on July i6th, 1945, lasted
three years and cost about half a billion dollars.

The inverse process, the fusion of nuclei into higher ones (e.g.
hydrogen into helium) is the source of energy of the sun and of all
stars. In the central parts of these the temperatures and pressures
are so high that the combination of four nucleons is possible in a
series of steps, through a chain reaction. The same has now been
accomplished here on earth by using a uranium bomb as ignition.
Thus we have now the H-bomb, which seemed to be an absolutely
hellish invention, as no method of abating the violence of the
explosion was known; but recently it has been announced that
ways of controlling this reaction have been found.

There is no doubt any more: all matter is unstable. If this were
not true the stars would not shine, there would be no heat and light
from the sun, no life on earth. Stability and life are incompatible.
Thus life is necessarily a dangerous adventure which may have a
happy end or a bad one. To-day the problem is how the greatest
adventure of the human race can be directed towards a happy end.

Now I wish to say a few words about the blessings which can be
obtained if men behave reasonably. There is, in the first line, the
problem of energy. When I was young, half a century ago, the time
our coal reserves would last was estimated to be a few hundred
years; petrol oil was not used then on a large scale. Meanwhile an
enormous amount of coal has been burnt, oil has been discovered
and used in an ever increasing rate. Yet the estimate of the duration
of the fossil fuel reserves is still many hundred years. Therefore it
seems not to be an urgent problem to find new sources of energy.

2l6 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

But this conclusion would be erroneous. Coal and oil are not only
sources of energy but the most important raw materials for in-
numerable chemical products. May I just mention the plastics
and their numerous applications. There will come a time when the
agricultural output does not suffice for feeding the ever-increasing
number of human beings. Then chemistry will be challenged to
produce substitutes, for which, of course, coal is the only available
raw material. Hence it is barbaric to burn coal and oil. Then the
social aspect of the question must not be forgotten. The day seems
to be not far away when in civilized countries no workmen will
be available who are willing to take up the dark and dangerous
profession of a miner, at least not for economically bearable wages.
England seems to approach this state of affairs already. Then there
are many countries which have neither coal nor oil; for these the
easily transportable nuclear fuel will be a blessing.

Another type of the peaceful applications of nuclear physics are
the radio-active by-products of atomic reactors. Instable, i.e.
radio-active isotopes of many elements are produced, which can
be applied to many purposes: as sources of radiation, instead of
the expensive radium, in medicine, technology, agriculture; for
instance for the treatment of cancer, the testing of materials, the
production of new species of plants through mutations. Perhaps
more important than all this is the idea of 'tracer elements'. By
adding a small amount of a radio-active isotope to a given element
it is possible to follow the fate of this element in chemical reactions,
even in living organisms, by observing the radiation emitted. An
ever increasing number of experiments in biological chemistry are
already using this method, which marks a new epoch in our know-
ledge of the processes of life.

All this, and what may develop from it in days to come, are
great things. An international conference at Geneva convened by
UNO has discussed the exploitation of all these possibilities by a
collaboration of all nations. I am not a nuclear physicist and have
not attended it. I hope the labours of this meeting will bring in a
rich harvest. But I cannot help asking: Can even a technical
paradise counterbalance the evil of the atomic bomb ?

I have used the phrase 'paradise on earth' already in the be-
ginning, but there I meant something different: not technical
progress, but the realization of the eternal yearning of Man for
'peace on earth 5 .

In regard to the opinions I wish to express now, I cannot rely on
my knowledge of physics, nor on my sporadic studies of history;
they seem to me just common sense, and they are shared by a

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 217

number of friends, leading scholars from different countries. We
believe that a major war between Great Powers there exist now
only two or three has become impossible, or at least will become
impossible in the near future. For it would lead, as I said already,
in all likelihood to general destruction, not only of the fighting
nations but also of the neutrals. GLAUSEWITZ' well known saying
that \var is the continuation of politics with other means does not
hold any more, for war has become insanity, and if the human race
is unable to renounce war its zoological name should not any longer
be derived from sapientia but from dementia.

The leading statesmen seem to be well aware of this situation.
The tuning down of the cold war which we are observing is an
indication that it is so. The fear of the enormity of the catastrophe
which might be the result of an armed conflict has led everywhere
to approaches and negotiations. But fear is a bad foundation for
reconciliation and solution of conflicts. Is it conceivable that the
peace resting on fear which we very likely are attaining at present
may be replaced by something better and more reliable?

I take it upon me if you regard me as a slightly ridiculous fellow
who refuses to acknowledge an awkward situation

Because, he argues trenchantly,
What must not happen cannot be,

as the grotesque philosopher PALMSTRdm says in the German poet
MORGENSTERN'S Songs from the Gallows.*

However, I am not alone with this view. EINSTEIN shared it and
has just before his death given a clear statement, together with
BERTRAND RUSSELL, the great philosopher, and others. A number
of 1 8 Nobel Laureates, chemists and physicists, gathered for a
scientific discussion at Lindau, have unanimously accepted a
declaration (the Mainau Statement) on similar lines. And many
other people and groups of people have published similar declara-
tions. May they appear to-day as dreamers: they are the builders
of the future world.

But not much time is available for their words to take effect. All
depends on this, the ability of our generation to re-adjust its thinking
to the new facts. If it is unable to do so then the days of civilized
life on earth are coming to an end. And even if all goes well, the
way will pass very, very close to the abyss.

* CHRISTIAN MORGENSTERN wrote deep and beautiful poetry which however
found little resonance in the public. Then he published several little volumes of
grotesque, apparently senseless verse under the title 'Galgenheder in which he
caricatured his philosophy through two strange figures, PALMSTROM and KORFF.
These books had a tremendous and lasting success.

2l8 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

For the world is full of conflicts appearing insoluble: Displaced
frontiers of countries; expelled populations; antagonism of races,
languages, national traditions, religions; the bankruptcy of the
colonial system; and finally the opposing economical ideologies,
capitalism and communism. Can we really hope that all these
terrible tensions will be solved without application of force?

Would it not be preferable, instead of the radical proposition to
abandon war, to make an attempt to prohibit the new weapons of
mass destruction by international agreement? This idea seems to
me (and my friends) impracticable for the following reasons.

The production of energy through nuclear reactions is already
being prepared and improved everywhere. A system of supervision
intended to inhibit the production of weapons of destruction can
function only in peace time. If war between major powers should
break out which might initially be conducted with conventional
weapons the supervision ceases. Is it reasonable to assume that a
nation in distress but believing that she could save herself with the
help of the atom bomb would be willing to renounce this last
resource even if she is liable to suffer badly herself?

Concerning those 'conventional weapons' I must confess that I
am unable to understand why they are not causing the same horror,
the same detestation which is generally felt to-day towards the
atomic weapons. They have ceased to be honest weapons used by
soldiers against soldiers and have become means of indiscriminate
destruction. They are not directed against military objects alone
but against the whole organization and productive capacity of the
enemy nation, against factories, railways, houses; they kill the
helpless, the old, children, women, they destroy the most noble and
valuable achievements of civilization, churches, schools, monu-
ments, museums, libraries, without any regard for historical
importance or irreparability. From the moral standpoint the
decisive step of warfare towards modern barbarism was the concept
of total war. Even without atomic weapons the prospect of the
effects of using ordinary bombs, in combination with chemical and
bacteriological poisons, is appalling enough.

Prohibition of atomic weapons alone is not justified, neither
morally nor by the actual facts. The human race can only be
saved by renouncing once for all the use of force through war.
To-day fear has produced such a precarious state of peace. The
next aim must be to stabilize this peace by strengthening the ethical
principles which secure the peaceful coexistence of men. CHRIST
has taught how man ought to behave towards man. The nations
have up to now acted and the Churches have not objected to this

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 21 9

attitude as if these commandments are valid only inside their
domains, but not in regard to their mutual relations. That is the
root of the evil. We can only survive if in the international sphere
distrust is replaced by understanding, jealousy by the will to help,
hatred by love. In our time, before our eyes, the doctrine of non-
violence has been victorious in the hands of a non- Christian,
MAHATMA GHANDI, who has liberated his country without bloodshed
(and I do not think that he would have acted differently if his
adversaries had not been the well-meaning British, but any other
nation). Why should it not be possible to follow his example?

I cannot make suggestions for the solution of the actual political
conflict. Yet I wish to discuss a few general points.

The first of these is that a tremendous number of men in all
countries have a personal interest in the preparation, and, if neces-
sary, the waging of war. There are big industries and many types
of business who make money from armaments. There are numerous
men who like the life of a soldier because of its romantic tradition,
or because they enjoy being rid of responsibility and having just
to obey. There are the officers: generals, admirals, air marshals,
etc., whose profession is war. They are still the advisers of present-
day governments. Finally there are the physicists, chemists, engineers
who invent new weapons and produce them. It would be an illusion
to make an attempt of stabilizing the present precarious peace
without taking any notice of all those people, without giving them
some substitute for the loss they have to expect. How this can be
done is beyond my competence except for one class of people, the
physicists whom I know well. Here I see no great difficulty.

One hears often hard words about the atomic physicists: all
calamity is the fault of these brain-athletes, not only the atom bomb
but also the bad weather. I have endeavoured to show that the
development of the human mind was bound to lead one day to the
disclosure and application of the energy stored in the atomic
nucleus. That this happened so quickly and so thoroughly as to
lead to a critical situation is the consequence of a tragic historical
accident: The discovery of the fission of uranium happened just
at the moment when HITLER acquired power, and just in that
country, where he acquired power. I, like many others, had then
to leave Germany and I have seen with my own eyes the panic
which struck the rest of the world when HITLER'S initial successes
made it appear possible that he might subjugate all nations of the
globe. The physicists emigrated from Central Europe knew that
there was no salvation if the Germans would succeed first to produce
the atomic bomb. Even EINSTEIN who had been a pacifist all his

22O DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

life shared this fear and was persuaded by some young Hungarian
physicists to warn President ROOSEVELT. Scholars emigrated from
Europe contributed much to the uranium project, the most prom-
inent of them ENRICO FERMI, perhaps the greatest experimental
physicist of our time next to RUTHERFORD. The direction of the
scheme remained in American hands. It seems to me that no blame
can be attached to the men who constructed the atom bomb unless
one accepts the teaching of extreme pacifism that power should
never be used even against the greatest evil. It is quite a different
matter with the application of the bombs against Japan in the last
phase of the war. I personally consider this to be a barbaric act,
and a foolish one. Responsible for it are not only politicians and
soldiers, but a small group of scientists who advised the deciding
committee appointed by President TRUMAN. One of these, FERMI,
has died meanwhile. Another, from reasons of conscience, has
given up all scientific activity, has become the head of a great
educational institution and works against the misuse of science.
Other members of this group have, as far as I know, not essentially
changed their life and activities, nor presumably their opinion about
the necessity of dropping the bombs on Japanese cities. If you wish
to get a glimpse of the psychology of the atomic physicists read
the clever and amusing book by LAURA FERMI, the widow of the
physicist, Atoms in the Family. The title of its last chapter is e A New
Toy, the Giant Cyclotron 3 . This word toy is significant, though
perhaps overdrawn. These men are swallowed up by their problems
and are triumphant if a solution is found, but ponder little about
the consequences of the results. And if they do so then with the
feeling: this is beyond our sphere of influence. The idea to abandon
research because its effects might be dangerous seems absurd to
them; for if they give up there would be plenty of others to con-
tinue, and in particular if the Americans were not on top, the Rus-
sians would be. And all, apart from a limited number, have after
the war returned to peaceful occupations, to research and teaching,
and they desire nothing better. Societies have been formed amongst
them to discuss and study the social responsibility of scientists and
to oppose the misuse of the discoveries.

There are of course a few physicists who have tasted power and
liked it, who are ambitious and want to preserve the influential
positions acquired during the war. But altogether I think that the
ideal of politics without force will be less resisted by scientists than
by other social groups. Even the ambitious and worldly scientists
will be satisfied by directing big projects of development and advising
the administrations of states in general politics. The consequences

DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE 221

of the appearance of this type of men for the development of
science itself are outside the frame of this discussion. May I be
allowed to express my personal opinion that from the standpoint
of fundamental research this development may turn out deplorable,
perhaps disastrous. The appearance of a new EINSTEIN is hardly
to be expected in such environments.

On the other hand, an admixture of scientists in politics and
administration seems to me an advantage because they are less
dogmatic and more open to argument than people trained in law
or classics. To illustrate this let me record a recent personal
experience.

There was the usual yearly gathering of Nobel Laureates, chemists
and a few physicists, at Lindau, Lake Konstanz, in July, for dis-
cussing scientific problems. OTTO HAHN, WERNER HEISENBERG and
myself submitted to them a declaration (called the Mainau State-
ment) prepared by us in collaboration with some other scholars of
different countries, in which the danger of the present situation
was emphasized and the abandonment of war demanded. Most of
the participants agreed at once, but a few had doubts. A famous
American scholar objected: I have just come from a visit to Israel
and convinced myself that the existence of this little nation can be
secured against the pressure of the Arabs only by the force of arms'.
That is plausible enough. But in the end he accepted our arguments
(the same as given here) and he signed the declaration with the
rest of us.

Exactly the same objection is made wherever the last wars have
left painful wounds, where boundaries have been shifted, popula-
tions expelled as in Israel, Korea, Indo-China, Germany.

I myself have experienced enough to know what it means to be
the victim of political persecution. I was allowed to return to my
home country Germany, but my proper home land Silesia, which
is now a part of Poland, is closed to me. That is a painful loss. But
fate has decided. To redress the situation by force is impossible
without much worse injustice and, very likely, general destruction.
We have to learn resignation, we have to practice understanding,
tolerance, the will to help instead of threats and force. Otherwise
the end of civilized man is near.

For I believe that BERTRAND RUSSELL is right if he never tires of
repeating: Our choice is only between Co-existence and Non-
existence. Let me end by quoting his words:

Tor countless ages the sun rose and set, the moon waxed and waned,
the stars shone in the night, but it was only with the coming of Man
that these things were understood. In the great world of astronomy

222 DEVELOPMENT AND ESSENCE OF THE ATOMIC AGE

and in the little world of the atom, Man has unveiled secrets which
might have been thought undiscoverable. In art and literature and
religion, some men have shown a sublimity of feeling, which makes
the species worth preserving. Is all this to end in trivial horror because
so few are able to think of Man rather than of this or that group of
men ? Is one race so destitute of wisdom, so incapable of impartial
love, so blind even to the simplest dictates of self-preservation, that
the last proof of its silly cleverness is to be the extermination of all
life on our planet? for it will be not only men who will perish, but
also the animals and plants, whom no one can accuse of communism
or anti-communism I cannot believe that this is to be the end.'

If we all refuse to believe this, and act accordingly, it will not be
the end.

A NEW YEAR'S MESSAGE

(From Physikalische Bldtter, vol. u, Jan. i, 1955.)

MUCH has changed in physics during the two decades I spent
abroad. It is no longer the quiet, pure science of old, but a
decisive factor in the power politics of nations. I have only been a
bystander of the revolution brought about by HAHN'S discovery of
uranium fission. It seems to me that the physicists of Germany
are not as conscious of this completely changed situation as those
of the Anglo-Saxon countries. There nobody can avoid the question
of conscience how far he wants to collaborate in the development
of forces which threaten the very existence of the civilized world.
I have often asked myself how Lord RUTHERFORD, the actual
founder of nuclear physics, would behave. He certainly was a
patriot and helped in the defence of his country during the First
World War. But he drew limits. When I came to Cambridge in
1933 FRITZ HABER was also there, ill and spiritually broken through
exile from his fatherland. I tried to bring him together with
RUTHERFORD; but he refused to shake hands with the originator
of chemical warfare. How would RUTHERFORD behave today?
He might have been able through the weight of his personality to
stop the unconditional surrender of means of destruction to politi-
cians and military. Some leading physicists of America have tried
just that, but without success. There is the document in which they
warned the American government not to use the atom bomb against
highly populated towns and in whi.ch they predicted correctly the
political and moral consequences it is known under the name of
the Franck Report after the chairman, my old friend JAMES FRANCK.
In America and England societies have been formed which aim
at solving the question of the social responsibility of the scientist.
As example I mention the American 'Society for Social Responsi-
bility in Science' (S.S.R.S.), of which I am a member. This
association informs its members by monthly news letters,* in these
we are told about discussions, talks, publications and books, and
given extracts from them, also statements are published by well
known men and women and finally letters from the readers are
printed. In the last number there are extracts from a letter by
ALBERT SCHWEITZER to the London Daily Herald about the hydrogen
bomb and also sentences from a lecture (Alex Wood Memorial

223

224 A NEW DEAR'S MESSAGE

Lecture 1954) by Professor KATHLEEN LONSDALE, the well-known
crystallographer, who became one of the first female members of
the Royal Society. She is a Quaker and a protagonist against the
misuse of scientific inventions for inhuman and political ends; she
is just back from a world trip via India and Japan to Australia
where she spread her ideas. She is a leader in English societies
which have similar aims.

As far as I know there is no such organization yet in Germany,
and that is only natural in view of the limitations which have been
placed on the German scientists by the occupation statute. But the
time has come when a new obligation arises from the lifting of this
restraint and with it the need for clarification of these problems.
It seems to me that the German Physical Society could be a forum
for such discussions. It is not by any means only a matter of the
most fundamental questions such as attitude towards war in
general and towards the use of means of destruction, which threaten
the existence of whole nations or even of all of civilized mankind.
But it is also a matter of the lesser and nevertheless important prob- '
lems which are concerned with the relation of the scientist to society.
To select a few points:

The threatening of freedom of science by military supervision
of research and censorship of publication, the spy witch-hunt as
it is now rampant in the United States, the founding of numerous
well-equipped state laboratories through which an increasing
number of scientists fall into dependence; finally the grave question
whether the successful researcher shall always remain only an
expert assistant or take a responsible part in important decisions.

German physics has achieved an enormous rebuilding of her
research and teaching materials in the few years since the collapse.
Let her use with equal verve the perhaps only short time between
now and complete freedom of action to clarify moral and social
questions which have been forced on the physicist in his role as
human being and citizen as a result of his own researches. If this
is left undone the freedom of science will be as greatly threatened
as the civic freedom of the individual scientist. And this problem
of responsibility is as international as science herself, A uniting of
the groups which discuss this in the different countries would
therefore be highly desirable.

FROM THE POSTSCRIPT TO "THE
RESTLESS UNIVERSE " (1951)

CONCLUSION

WE have reached the end of our journey into the depth of matter.
We have sought for firm ground and found none. The deeper
we penetrate, the more restless becomes the universe, and the
vaguer and cloudier. It is said that ARCHIMEDES, full of pride in
his machines, cried, e Give me a place to stand, and I will move
the world!' There is no fixed place in the Universe: all is rushing
about and vibrating in a wild dance. But not for that reason only
is ARCHIMEDES' saying pontifical. To move the world would mean
contravening its laws; but these are strict and invariable.

The scientist's urge to investigate, like the faith of the devout or
the inspiration of the artist, is an expression of mankind's longing
for something fixed, something at rest in the universal whirl: God,
Beauty, Truth.

Truth is what the scientist aims at. He finds nothing at rest,
nothing enduring, in the universe. Not everything is knowable,
still less predictable. But the mind of man is capable of grasping
and understanding at least a part of Creation; amid the flight of
phenomena stands the immutable pole of law.

So schafF ich am sausenden Webstuhl der Zeit
Und wirke der Gottheit lebendiges Kleid.

GOETHE, Faust.

'Tis thus at the roaring Loom of Time I ply,

And weave for God the Garment thou seest Him by.

(CARLYLE'S translation.)

POSTSCRIPT

Since I wrote the last lines, 15 years ago, great and formidable
events have happened. The dance of atoms, electrons and nuclei,
which in all its fury is subject to God's eternal laws, has been
entangled with another restless Universe which may well be the
Devil's: the human struggle for power and domination, which
eventually becomes history. My optimistic enthusiasm about the
disinterested search for truth has been severely shaken. I wonder
at my simplemindedness when I re-read what I said on the modern
fulfilment of the alchemists 5 dream:

225

226 FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE" (1951)

'Now however, the motive is not the lust for gold, cloaked by the
mystery of magic arts, but the scientists' pure curiosity. For it is
clear from the beginning that we may not expect wealth too.'

Gold means power, power to rule and to have a big share in the
riches of this world. Modern alchemy is even a short-cut to this
end, it provides power directly; a power to dominate and to
threaten and hurt on a scale never heard of before. And this power
we have actually seen displayed in ruthless acts of warfare, in the
devastation of whole cities and the destruction of their population.
Such acts, of course, have been achieved by other means. In the
same war other cities than Hiroshima, with a considerable percentage
of their population, have been destroyed a little slower by ordinary
explosives. Every previous war had its technical 'progress' in
destruction, back to the stone age when the first bronze weapons
conquered flint axes and arrow heads. Still there is a difference.
Many states, populations, civilizations have perished through
superior power, but there were vast regions unaffected and room
was left for new growth. To-day the globe has become small, and the
human race is confronted with the possibility of final self-destruction.

When the question of a new edition of this book arose I felt a
considerable embarrassment. To bring it up-to-date I had to
write an account of the scientific development since 1935. But
although this period is as full of fascinating discoveries, ideas,
theories, as any previous epoch, I could not possibly describe them
in the same tone in which the book was written; namely, in the
belief that a deep insight into the workshop of nature was the first
step towards a rational philosophy and to worldly wisdom. It seems
to me that the scientists who led the way to the atomic bomb were
extremely skilful and ingenious, but not wise men. They delivered
the fruits of their discoveries unconditionally into the hands of
politicians and soldiers; thus they lost their moral innocence and
their intellectual freedom.

-On July 1 6, 1945, the first experimental bomb exploded near
Los Alamos, New Mexico. This was certainly one of the greatest
triumphs of theoretical physics if measured not by the subtlety of
ideas but by the effort made in money, scientific collaboration and
industrial organization. No preliminary experiment was possible,
the tremendous risk was taken in the confidence that the theoretical
calculations based on laboratory experiments were accurate. There-
fore it is no wonder that the physicists who watched the terrific
phenomenon of the first nuclear explosion felt proud and relieved
from a heavy responsibility. They had done a great service to their
country and to the community of allied nations.

FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE" (1951) 227

But when, a few weeks later, two 'atomic bombs 5 were dropped
over Japan and destroyed the crowded cities of Hiroshima and
Nagasaki, they discovered that a more fundamental responsibility
was on their shoulders.

The world had become pretty callous against the horrors of the
war. HITLER'S seed had grown. His was the idea of total war, and
his bombs smashed Rotterdam and Coventry. But he found keen
pupils. In the end the bombers of both sides succeeded in a sys-
tematic devastation of Central Europe. A great part of its historic
and artistic treasures, the inheritance of thousands of years went up
in flames. An architectural jewel like Dresden was destroyed in
one of the last days of the European war, and 100,000 civilians, men,,
women and children, are said to have perished with it. I do not
doubt that those responsible for this act can rightfully claim tactical
and strategical necessity; and the world in general found sufficient
justifications, ranging from blind hatred and the wish of retribution
to the quasi-humane idea that to shorten the war all means are
good enough. Ethical standards had fallen sharply, indeed.

Still the two atomic bombs dropped on Japan made a stir, and
when details of the human tragedy became known there was
something like an awakening of conscience in many parts of the
world.

.This is not the place to express my personal judgment of the
statesmen who decided to use this brutal application of power.
Cases of precedence are plentiful there is not much difference in
the responsibility for killing 20,000 in one night or 50,000 in one
minute. But being a scientist I am concerned with the question of
how far science and scientists share the responsibility.

The motives of those who took part in the development of nuclear
explosives were certainly above reproach : Many of them were just
drafted to this work as their war service, others joined it, driven by
the apprehension that the Germans might produce the bomb first.
Yet there was no organization of scientists which could form a
general opinion. Single men became little cog-wheels in the
tremendous machine, which was directed by political and military
authorities. The leading physicists became scientific advisers of
these authorities and experienced the new sensation of power and
influence. They enjoyed their work and its tremendous success^
and forgot for the time being to think hard about its consequences.
It is true that a group of scientists warned the U.S. Government
not to use the bomb against cities, but to demonstrate its existence
and power in a less murderous way, for instance on the top of
Fujiyama mountain. They predicted very accurately the disastrous

228 FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE 9 ' (1951)

political consequences which an attack on a city would have. But
their advice was neglected.

The principal discrepancy between public opinion in the United
States and the conviction of the scientists is concerned with secrecy.
The scientists are convinced that there is no secret in science.
There may be technical tricks which can be kept secret for a limited
period. But the laws of nature are open to anybody who is trained
in using the scientific method of research.

Therefore it was futile to keep the atomic bomb project from
being known to the Russian allies, and the maintenance of this
secret has with necessity transformed them from old friends into
enemies. They felt menaced by a tremendous new weapon; they
started to develop it themselves, and they obtained it in a shorter
time than was ever expected.

On the other hand this phantom of secrecy had disastrous effects
on the development of nuclear physics in America. Many physicists
have been subjected to suspicion and even to accusation of dis-
loyalty. The whole of science has been hampered by the classifica-
tion of discoveries into secret and open ones, and by the supervision
of publication. There is no doubt that certain security measures,
mainly in regard to technical questions, are unavoidable. But the
subordination of fundamental research to political and military
authorities is detrimental. The scientists themselves have learned
"by now that the period of unrestricted individualism in research
has come to an end. They know that even the most abstract and
remote ideas may one day become of great practical importance
like EINSTEIN'S law of equivalence of mass and energy. They have
begun to organize themselves and to discuss the problem of their
responsibility to human society. It would be left to these organiza-
tions to find a way to harmonise the security of the nations with the
freedom of research and publication without which science must
stagnate.

The release of nuclear energy is an event comparable to the first
fire kindled by prehistoric man though there is no modern
Prometheus but teams of clever yet less heroic fellows, useless as
inspiration for epic poetry. Many believe that the new discoveries
may lead either to immense progress or to equal catastrophe, io
paradise or to hell. I, however, think that this earth will remain
what it always was; a mixture of heaven and hell, a battlefield of
angels and devils. Let us have a look around: what are the
prospects of this battle, and what can we do to help the good cause?

To begin with the devil's part, there i$ the hydrogen bomb. We
have seen that, though almost all matter is unstable in principle.

FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE" (1951) 22

we are protected against nuclear catastrophe by the low temperatures
on earth, which even in our hottest furnaces are quite insufficient
to initiate nuclear fusion. But the discovery of fission has destroyed
this security. The temperature in an exploding uranium bomb is
presumably high enough to start the fusion of hydrogen with the
help of the e carbon cycle 5 , which is the source of stellar energy, or a
similar catalytic process. Thus an explosive of many thousand times
higher efficiency than the fission bomb could be made from a
material available in abundance. Of course, work has started with
the usual argumentation: if we do not do it, the other fellow (mean-
ing the Russian) will. If it succeeds there will be a new instrument
of wholesale destruction, but no peaceful application of the new
forces seems to be possible. No way is known to slow down fusion
in order to use it as a fuel. A perfectly hellish prospect.

Fission however has many and far-reaching applications of a
peaceful kind. It can be used as fuel, since the reaction velocity can
be controlled. Each pile produces an enormous amount of heat
which at present is wasted in most cases. Power stations using
uranium or thorium as fuel are possible, as the difficulties connected
with the pernicious radiation could certainly be overcome. The
question is however an economic one. The raw material is rare,
and if the same amount of energy which is at present made from
coal would be produced by nuclear reactors, the whole uranium
ore at present or in future available would be used up in less than
half a century. Hence it is improbable that the new fuel will be
able to compete with coal and oil. Under certain conditions, how-
ever, this may be the case, namely where the advantage of the small
bulk and weight of nuclear fuel, as compared with that of coal or
oil, is decisive. There is a possibility of increasing the efficiency of
fission by 'breeding', i.e. by directing the process in a pile in such
a way that a great proportion of the nuclei present is transformed
into fissionable isotopes. This would mean an extension of the raw
material over a much longer period.

Apart from the still problematic application of nuclear reactions
for power production, there are numerous others which have already
led to great progress'and which are more promising. There is first
the generation of new isotopes in the pile. Our knowledge of the
stability of nuclei and of the laws of their interaction has been
immensely increased. Some of the radio-active products can be
used in medicine for therapeutical purposes, replacing for instance
radium in the fight against cancer. The most important application
is the so-called 'tracer method' which is revolutionizing chemistry
and biology. Already in the first period of radio-activity v. HEVESY

230 FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE" (I95 1 )

had the idea to trace the fate of atoms in chemical or biological
processes by adding to them a small amount of a radio-active
isotope. This discloses its presence by radiation, and as the methods
of detection of radiation are extremely sensitive, one can thus deter-
mine much smaller amounts of an element than with the balance.
It is even possible to investigate the distribution of atoms in living
tissue. The actual application of this idea was formerly restricted
to the few atomic types for which naturally radio-active isotopes
were known. Isotopes are now available for almost all elements of
the periodic system. The work on this line, though hardly begun,
has already led to important results, and will lead to still more.

But what are these important results compared with the spectre
lurking in the background, the possibility of atomic warfare on a
great scale?

In combination with other infernal contraptions, like rockets to
deliver bombs at large distances, chemical, biological and radio-
active poisons, such a war must mean a degree of human suffering
and degradation which is beyond the power of imagination. No
country would be immune, but those with highly developed industry
would suffer most. It is very doubtful whether our technological
civilization would survive such a catastrophe. One may be inclined
to regard this as no great loss, but as a just punishment for its
shortcomings and sins: the lack of productive genius in art and
literature, the neglect of the moral teachings of religion and
philosophy, the slowness to abandon outdated political conceptions,
like national sovereignty. Yet we are all involved in this tragedy,
and the instinct of self-preservation, the love of our children, makes
us think about a way of salvation.

There are the two political colossi, U.S.A. and U.S.S.R., both
pretending to aim at nothing but peace, but both rearming with all
their power to defend their ideology and way of life, and between
them is a weak and divided Europe, trying to steer a middle course.
Both sides are greedily devouring the latest achievements of
scientific technology for their armed forces. Both have some kind
of theory for .their way of life in which they believe with an amazing
fanaticism. Yet the foundations of these theories are rather doubtful.
They use the same words for different or even opposite ideas, as for
instance 'democracy*, which in the West means a system of parlia-
mentary representation freely and secretly elected, but in the East
means something quite different and hard to formulate (a compli-
cated economic and political pyramid of bureaucracy which aims
at representing, and working for, e the people') . In other ways the
American theory is much vaguer than the Russian, and that seems

FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE " (1951) 231

to have a historical reason. America has grown by expansion in a
practical vacuum; the pioneers of the West had to overcome terrific
natural obstacles, but negligible human resistance. The Russia of
today had to conquer not only natural but human difficulties: she
had to break up the rotten system of the Czars and to assimilate
backward Asiatic tribes; now she has set herself the task of bringing
her brand of modernization to the ancient civilizations of the Far
East. For this purpose it is indispensable to have a well-defined
doctrine full of slogans, which appeals to the needs and instincts
of the poverty-stricken masses. Thus one understands the power
which MAJRX'S philosophy has gained in the East. What can we
scientists do in this conflict? We can join the spiritual, religious,
philosophical forces, which reject war on ethical grounds. We can
even attack the ideological foundations of the conflict itself. For
science is not only the basis of technology but also the material
for a sound philosophy. And the development of modern physics
has enriched our thinking by a new principle of fundamental
importance, the idea of complementarity. The fact that in an
exact science like physics there are found mutually exclusive and
complementary situations which cannot be described by the same
concepts but need two kinds of expressions, can be applied to other
fields of human activity and thought. Some such applications to
biology and psychology were suggested by NIELS BOHR. In philo-
sophy there is the ancient and central problem of free will. Any
act of willing can be regarded on the one side as a spontaneous
process in the conscious mind, on the other as a product of motives
depending on past or present impressions from the outside world.
If one assumes that the latter are subject to deterministic laws of
nature, one has a conflict between the feeling of freedom of action
and the necessity of a natural process. But if one regards this as
an example of complementarity the apparent contradiction turns
out to be nothing but an epistemological error. This is a healthy
way of thinking, which properly applied may remove many violent
disputes not only in philosophy but in all ways of life: for instance
in politics.

Marxian philosophy, which is a hundred years old, knows of
course nothing of this new principle. However, a prominent
Russian scientist has recently attempted to interpret it from the
standpoint of 'dialectic materialism', which teaches that all thinking
consists of a thesis opposed by an antithesis; after some struggle,
they are combined in a synthesis. In this Marxian dogma, so he
claims, you have the prediction of what has happened in physics,
for instance in optics: NEWTON'S thesis that light consists of particles

232 FROM THE POSTSCRIPT TO "THE RESTLESS UNIVERSE*' (1951)

was opposed by HUYGENS' antithesis that it consists of waves, until
both were united in the synthesis of quantum mechanics. That is
all very well and indisputable, though a little trivial. But why not
go further and apply it to the two competing ideologies: Liberalism
(or Capitalism) and Communism, as thesis and anti- thesis ? Then
one would expect a synthesis of some kind, instead of the Marxian
doctrine of the complete and permanent victory of communism.
It can hardly be expected that the ideas of MARX, developed about
100 years ago, can throw much light on the development of modern
science. The opposite is more likely: that the new philosophical
ideas developed by science during these 100 years may help towards
a deeper understanding of social and political relations. Indeed,
we find two systems of thought which deal with the same structure,
the state, in completely different, apparently contradictory ways.
One starts from the freedom of the individual as the basic concep-
tion, the other from the collective interest of the community.

This distinction corresponds roughly to the two aspects of the
problem of willing which we have just mentioned: the subjective
feeling of freedom on the one hand, the causal chain of motives on
the other. Thus the West idealises political and economical
liberalism, the East collective life regulated by an all-powerful
state. But as it seems likely that the contradiction in the problem
of free will can be solved by applying the idea of complementarity,
the same will hold for the contradiction of political ideologies.
Thus the intellectual gulf between West and East may be bridged,
and that is the service which natural philosophy can offer in the
present crisis.

The world which is so ready to use the gifts of science for mass
destruction would do well to listen to this message of reconciliation
and co-operation.